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@@ -105,837 +110,1813 @@ %% ───────────────────────────────────────────────────────────────────────────── \begin{document} -\begin{titlepage} - \centering - \vspace*{2cm} - {\Huge\bfseries\sffamily\color{optblue} Mathematical Foundations\par} - \vspace{0.6cm} - {\large\sffamily Optimiz-rs — High-Performance Optimization in Rust\par} - \vspace{0.4cm} - {\large\sffamily\color{gray} Stochastic Processes · Jump Processes · Optimal Control\\ - Mean Field Games · Information Theory · Differential Geometry\par} - \vspace{1cm} - \textcolor{optgold}{\rule{0.5\linewidth}{1.5pt}} - \vspace{1cm} +\section{Mathematical Foundations}\label{mathematical-foundations} - \begin{tabular}{rl} - \textbf{Version} & 0.1 \\ - \textbf{Date} & \today \\ - \textbf{Docs} & \url{https://optimiz-r.readthedocs.io} \\ - \textbf{Repo} & \url{https://github.com/ThotDjehuty/optimiz-rs} \\ - \end{tabular} +This page develops the core mathematics underlying Optimiz-rs's Rust kernels --- from first principles through advanced theory. Each section opens with a \textbf{definition block}, builds intuition through \textbf{examples and diagrams}, and closes with a \textbf{notebook micro-check}. For complete walkthroughs see \texttt{examples/notebooks/}. - \vfill - \textit{This document is auto-generated from \texttt{docs/source/theory/mathematical\_foundations.md}.} -\end{titlepage} +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\tableofcontents -\newpage +\subsection{1 · Differential Evolution (DE)}\label{differential-evolution-de} -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Differential Evolution (DE)} -%% ═══════════════════════════════════════════════════════════════════════════ +\subsubsection{Background}\label{background} -\subsection{Background} +DE is a gradient-free population-based optimizer for \(f: \mathbb{R}^d \to \mathbb{R}\), not required to be smooth or convex. At generation \(g\) we maintain \(N\) candidate solutions \(\{\mathbf{x}_{i,g}\} \subset \mathbb{R}^d\). -DE is a gradient-free population-based optimizer for $f: \R^d \to \R$, -not required to be smooth or convex. At generation $g$ we maintain $N$ candidate -solutions $\{\mathbf{x}_{i,g}\} \subset \R^d$. +\textbf{Key insight:} The difference vector \(\mathbf{x}_{r_2}-\mathbf{x}_{r_3}\) is an unbiased directional finite-difference of \(f\), so DE implicitly estimates curvature without Jacobians. -\textbf{Key insight:} The difference vector $\mathbf{x}_{r_2}-\mathbf{x}_{r_3}$ is an -unbiased directional finite-difference of $f$, so DE implicitly estimates curvature -without Jacobians. +\subsubsection{\texorpdfstring{1.1 Geometric Intuition --- Mutation in \(\mathbb{R}^2\)}{1.1 Geometric Intuition --- Mutation in \textbackslash mathbb\{R\}\^{}2}}\label{geometric-intuition-mutation-in-mathbbr2} -\subsection{Operators} +\begin{verbatim} + x_r3 * + | + | F*(x_r2 - x_r3) F = scale factor in [0,2] + | ------------------> + x_r2 * * v_i (mutant) + \\ / + \\________________/ + (difference vec) -\begin{center} -\begin{tabular}{L{3cm} L{7cm} L{3cm}} -\toprule -\textbf{Step} & \textbf{Formula} & \textbf{Role} \\ -\midrule -Mutation (rand/1) & - $\mathbf{v}_{i,g} = \mathbf{x}_{r_1} + F(\mathbf{x}_{r_2}-\mathbf{x}_{r_3})$ & - Explore \\[3pt] -Binomial crossover & - $u_{i,j} = v_{i,j}$ if $U(0,1) v_i + base vector mutation vector added +\end{verbatim} -\textbf{Convergence (informal):} Under bounded population diversity and Lipschitz $f$, the -best-so-far value converges a.s.\ to a stationary point as $N,g\to\infty$ -(Price et al.\ 2005). - -\subsection{Self-Adaptive jDE (default)} - -Parameters $F,CR$ are per-individual and reset stochastically each generation: -\begin{equation} - F_i^{g+1} = \begin{cases} - F_{\min} + r_1 F_{\max} & r_2 < \tau_1,\\ - F_i^g & \text{otherwise,} - \end{cases} - \qquad - CR_i^{g+1} = \begin{cases} - U(0,1) & r_3 < \tau_2,\\ - CR_i^g & \text{otherwise.} - \end{cases} -\end{equation} - -$\tau_1=\tau_2=0.1$ by default. On rugged landscapes this produces bimodal $F$ -histograms concentrated near~$0.8$ — a sign the landscape is highly multimodal. - -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Stochastic Processes} -%% ═══════════════════════════════════════════════════════════════════════════ - -These form the probabilistic backbone of all continuous-time models in Optimiz-rs. - -\subsection{Brownian Motion} - -\begin{defbox}{Wiener Process} -A stochastic process $W = (W_t)_{t\ge 0}$ on $(\Omega,\mathcal{F},\PP)$ -is a \emph{standard Brownian motion} if: -\begin{enumerate} - \item $W_0 = 0$ a.s. - \item Increments are \textbf{independent}: $W_t - W_s \perp \mathcal{F}_s$ for $t>s$. - \item $W_t - W_s \sim \N(0, t-s)$ for all $0\le s2$). - \item \textbf{Self-similarity:} $c^{-1/2}W_{ct} \overset{d}{=} W_t$ (Hurst exponent $H=\tfrac12$). + +\item + \(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3\) are three \textbf{distinct} randomly selected parents. +\item + The mutant \(\mathbf{v}_i\) lands on the other side relative to \(\mathbf{x}_{r_1}\). +\item + \textbf{Crossover} then mixes \(\mathbf{v}_i\) and \(\mathbf{x}_i\) dimension-by-dimension with probability \(CR\), producing trial vector \(\mathbf{u}_i\). +\item + \textbf{Selection} keeps \(\mathbf{u}_i\) only if it improves over \(\mathbf{x}_i\) --- pure greedy. \end{itemize} -\textbf{Example — Geometric BM:} -$S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)$ -is the Black--Scholes price model (log-normal marginals, nowhere-differentiable paths). +\subsubsection{1.2 Operators}\label{operators} -\subsection{Itô Calculus} +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2857}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.4286}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2857}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Step +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Formula +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Role +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Mutation (rand/1) & \(\mathbf{v}_{i,g} = \mathbf{x}_{r_1} + F(\mathbf{x}_{r_2}-\mathbf{x}_{r_3})\) & explore \\ +Binomial crossover & \(u_{i,j} = v_{i,j}\) if \(U(0,1) landscape is multimodal (increase $N$). + Collapsed near 0 -> diversity loss; restart with random perturbation. +2. **Check $CR$ distribution.** Uniform -> dimensions not interacting. + Collapsed near 0 -> DE treating dimensions independently (separable function). +3. **Increase $N$** to $\approx 10d$ for $d > 20$. +\end{tcolorbox} + +\textbf{Notebook check} (\texttt{05\_performance\_benchmarks.ipynb}): Plot \(F_i, CR_i\) histograms every 50 generations; expect values clustering in \([0.5,0.9]\) on hard problems. + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{2 · Stochastic Processes}\label{stochastic-processes} + +These form the probabilistic backbone of all continuous-time models in Optimiz-rs. We build the theory from scratch: random walk → Brownian motion → Itō calculus → SDEs → jump-diffusions. + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsubsection{2.1 Brownian Motion}\label{brownian-motion} + +\paragraph{2.1.0 Intuitive Construction --- From Random Walk to BM}\label{intuitive-construction-from-random-walk-to-bm} + +\textbf{Step 1 --- Discrete random walk.} Flip a fair coin \(n\) times per unit time. Define \(\xi_k = +1\) (heads) or \(-1\) (tails) i.i.d. After \(n\) steps of size \(1/\sqrt{n}\): + +\[S^{(n)}_t = \frac{1}{\sqrt{n}}\sum_{k=1}^{\lfloor nt \rfloor} \xi_k.\] + +By the \textbf{Central Limit Theorem}, as \(n\to\infty\): \(S^{(n)}_t \xrightarrow{d} W_t \sim \mathcal{N}(0,t)\). + +\begin{verbatim} +Coin-flip random walk (n=20 steps per unit time): + + W_t ++2 | * * + | * * * + 0 | * * * * * + |* * * +-2 | * + +----------------------------> t + 0 0.5 1.0 + + n → ∞ ("zoom out"): jagged → smooth BM fan +\end{verbatim} + +\textbf{Step 2 --- Scaling limit.} The normalization \(1/\sqrt{n}\) is crucial: - Without it, variance grows as \(n\) (diverges). - With \(n^{-1/2}\): variance = \(n \cdot (1/\sqrt{n})^2 \cdot t = t\) --- exactly right. + +This is why \(W_t \sim \mathcal{N}(0,t)\): \textbf{variance accumulates linearly in time}. + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Wiener Process},breakable] +A stochastic process $W = (W_t)_{t\ge 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$ +is a *standard Brownian motion* if: + +1. $W_0 = 0$ a.s. +2. Increments are **independent**: $W_t - W_s \perp \mathcal{F}_s$ for $t>s$. +3. $W_t - W_s \sim \mathcal{N}(0, t-s)$ for all $0\le s{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3000}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3667}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Property +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Formula +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Intuition +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Mean & \(\mathbb{E}[W_t] = 0\) & No drift --- symmetric random walk \\ +Variance & \(\operatorname{Var}(W_t) = t\) & Uncertainty grows with time \\ +Covariance & \(\operatorname{Cov}(W_s,W_t) = \min(s,t)\) & Shared history up to first time \\ +Non-differentiability & \(\lim_{h\to 0}(W_{t+h}-W_t)/h\) diverges a.s. & Too ``rough'' for ordinary calculus \\ +Quadratic variation & \([W]_T = T\) & Core source of Itō correction term \\ +Self-similarity & \(c^{-1/2}W_{ct} \overset{d}{=} W_t\) & Fractal structure, Hurst \(H=\tfrac12\) \\ +\end{longtable} +} + +\textbf{Quadratic variation derivation (step by step):} + +Partition \([0,T]\) into \(n\) pieces of width \(\Delta = T/n\). Sum of squared increments: + +\[\sum_{k=0}^{n-1}(W_{t_{k+1}}-W_{t_k})^2 \overset{?}{=} T \quad \text{as } n\to\infty.\] + +\textbf{Step 1} --- Each increment: \((W_{t_{k+1}}-W_{t_k})^2 \sim \Delta \cdot \chi_1^2\), so \(\mathbb{E}[(W_{t_{k+1}}-W_{t_k})^2] = \Delta\). + +\textbf{Step 2} --- Sum of means: \(\sum_{k=0}^{n-1} \Delta = n\Delta = T\). + +\textbf{Step 3} --- Variance of the sum: \(\operatorname{Var}\!\left(\sum (W_{t_{k+1}}-W_{t_k})^2\right) = n \cdot 2\Delta^2 = 2T^2/n \xrightarrow{n\to\infty} 0\). + +\textbf{Conclusion:} \(\sum (W_{t_{k+1}}-W_{t_k})^2 \xrightarrow{L^2} T\). We write \(dW_t^2 = dt\).\\ +This \textbf{single identity} is the engine of all Itō calculus. + +\begin{tcolorbox}[colback=thmbg!60,colframe=optgold,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Theorem / Key Idea}} — Why dW² = dt is remarkable},breakable] +In ordinary calculus, $dx^2 \approx dx \cdot dx \to 0$ (second-order infinitesimal). +For Brownian motion, $(dW)^2 = dt$ is **first-order** — it does **not** vanish! + +Physically: BM paths oscillate so rapidly ($\sim t^{0.5}$ scale) that their squared +increments accumulate at rate $1$ — comparable to the drift $dt$. + +This is the **only** reason Itō's lemma has an extra term. +\end{tcolorbox} + +\textbf{Multiple sample paths} --- the fan widens as \(\propto\sqrt{t}\): + +\begin{verbatim} +W_t + +2 | ........... + | .... ..... + 0 |.. ..... <- E[W_t] = 0 (all paths centered) + | ....--...... + -2 | ........ + +--------------------------> t + 0 T + + Width ~ 2*sqrt(t) (95% of paths stay within +/- 2*sqrt(t)) + + Each dot is a BM path — different because different coin flips. + As T grows, the "trumpet" opens wider. +\end{verbatim} + +\textbf{Example --- Geometric BM:} \(S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)\) is the Black-Scholes price model. Log-normal marginals; continuous, nowhere-differentiable paths: + +\begin{verbatim} +S_t +| .---. +| .--./ \----. +| / \---------. +|/ ++-------------------------------> t + 0 T +\end{verbatim} + +\textbf{Multiple sample paths} --- the fan widens as \(\propto\sqrt{t}\): + +\begin{verbatim} +W_t + +2 | ........... + | .... ..... + 0 |.. ..... <- E[W_t] = 0 (all paths centered) + | ....--...... + -2 | ........ + +--------------------------> t + 0 T + + Width grows as sqrt(t) (68% of paths stay within +/- sqrt(t)) +\end{verbatim} + +\textbf{Example --- Geometric BM:} \(S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)\) is the Black-Scholes price model. Log-normal marginals; continuous, nowhere-differentiable paths: + +\begin{verbatim} +S_t +| .---. +| .--./ \\----. +| / \\---------. +|/ ++-------------------------------> t + 0 T +\end{verbatim} + +\subsubsection{2.2 Itō Calculus}\label{itux14d-calculus} + +\paragraph{2.2.0 Why You Cannot Use Ordinary Integration}\label{why-you-cannot-use-ordinary-integration} + +Attempt to define \(\int_0^T W_t\,dW_t\) using a Riemann sum: pick \(W_{t_k}\) at the \textbf{left endpoint} → get one answer; pick \((W_{t_k}+W_{t_{k+1}})/2\) (midpoint) → get a \emph{different} answer. + +This ambiguity occurs because \(W\) is not of bounded variation. \textbf{Itō's convention} (left endpoint) is the only one that produces a \textbf{martingale} --- ensuring no look-ahead. + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Itō Integral},breakable] +For adapted $f \in \mathcal{L}^2$ (i.e. $\mathbb{E}\!\int_0^T f_t^2\,dt < \infty$): + +$$\int_0^T f_t\,dW_t \;:=\; L^2\text{-}\lim_{|\pi|\to 0} \sum_{k} f_{t_k}(W_{t_{k+1}}-W_{t_k}).$$ + +Key guarantees: +- **Zero mean:** $\mathbb{E}\!\left[\int_0^T f_t\,dW_t\right] = 0$. +- **Itō isometry:** $\mathbb{E}\!\left[\left(\int_0^T f_t\,dW_t\right)^2\right] = \mathbb{E}\!\int_0^T f_t^2\,dt$. +- **Martingale:** $M_t = \int_0^t f_s\,dW_s$ satisfies $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$. +\end{tcolorbox} + +\textbf{Itō isometry --- proof sketch:}\\ +Let \(I_T = \sum_k f_{t_k}\Delta W_k\) (simple process). Then: + +\[\mathbb{E}[I_T^2] = \sum_{j,k}\underbrace{\mathbb{E}[f_{t_j}\Delta W_j \cdot f_{t_k}\Delta W_k]}_{\text{cross terms}}\] + +For \(j \neq k\) (say \(j < k\)): \(f_{t_j}\Delta W_j\) and \(f_{t_k}\) are both \(\mathcal{F}_{t_k}\)-measurable, while \(\Delta W_k\) is \textbf{independent} of \(\mathcal{F}_{t_k}\) with mean 0 → cross term \(= 0\). + +For \(j = k\): \(\mathbb{E}[f_{t_j}^2 (\Delta W_j)^2] = \mathbb{E}[f_{t_j}^2]\Delta t_j\) (independence of \(f_{t_j}\) and \(\Delta W_j\)). + +\[\Rightarrow \mathbb{E}[I_T^2] = \sum_k \mathbb{E}[f_{t_k}^2]\Delta t_k \xrightarrow{|\pi|\to 0} \mathbb{E}\int_0^T f_t^2\,dt. \quad \checkmark\] + +\paragraph{2.2.1 Itō's Lemma --- Full Derivation}\label{itux14ds-lemma-full-derivation} + +\begin{tcolorbox}[colback=thmbg!60,colframe=optgold,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Theorem / Key Idea}} — Theorem — Itō's Lemma},breakable] +For $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and $F \in C^{1,2}([0,T]\times\mathbb{R})$: + +$$\boxed{dF(t,X_t) = \partial_t F\,dt + \partial_x F\,dX_t + \tfrac{1}{2}\sigma_t^2\,\partial_{xx}F\,dt}$$ + +Expanded: + +$$dF = \underbrace{\left(\partial_t F + \mu_t\,\partial_x F + \tfrac12\sigma_t^2\,\partial_{xx}F\right)}_{\text{drift}}\,dt ++ \underbrace{\sigma_t\,\partial_x F}_{\text{diffusion}}\,dW_t.$$ +\end{tcolorbox} + +\textbf{Derivation --- Taylor expand \(F(t+dt, X_{t+dt})\):} + +\[dF = \partial_t F\,dt + \partial_x F\,dX + \tfrac12\partial_{xx}F\,(dX)^2 + \underbrace{\partial_{tx}F\,dt\,dX + \ldots}_{\to 0} \] + +Compute \((dX)^2\) using the \textbf{Itō multiplication table}: + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{}lll@{}} +\toprule\noalign{} +× & \(dt\) & \(dW_t\) \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +\(dt\) & \(0\) & \(0\) \\ +\(dW_t\) & \(0\) & \(dt\) \\ +\end{longtable} +} + +\[\begin{aligned} +(dX_t)^2 &= (\mu_t\,dt + \sigma_t\,dW_t)^2 \\ + &= \mu_t^2\underbrace{(dt)^2}_{0} + 2\mu_t\sigma_t\underbrace{dt\cdot dW_t}_{0} + \sigma_t^2\underbrace{(dW_t)^2}_{dt}\\ + &= \sigma_t^2\,dt. +\end{aligned}\] + +Substituting: + +\[dF = \partial_t F\,dt + \partial_x F(\mu_t\,dt + \sigma_t\,dW_t) + \tfrac12\partial_{xx}F\cdot\sigma_t^2\,dt\] + +\[= \left(\partial_t F + \mu_t\partial_x F + \tfrac12\sigma_t^2\partial_{xx}F\right)dt + \sigma_t\partial_x F\,dW_t. \quad \checkmark\] + +\textbf{The extra term \(\tfrac12\sigma^2\partial_{xx}F\,dt\) is the ``Itō correction''.}\\ +In ordinary calculus \((dW)^2=0\), so it vanishes. In stochastic calculus, BM oscillates so rapidly that \((dW)^2 = dt\) --- a first-order effect. + +\textbf{Multidimensional version} (for vector \(\mathbf{X}\in\mathbb{R}^n\), matrix noise): + +\[dF = \partial_t F\,dt + \sum_i \partial_{x_i}F\,dX_i + \tfrac12\sum_{i,j}\partial_{x_ix_j}F\,d[X_i,X_j]_t\] + +where \(d[X_i, X_j]_t = d\langle X_i, X_j\rangle_t\) is the quadratic co-variation. + +\paragraph{2.2.2 Worked Examples of Itō's Lemma}\label{worked-examples-of-itux14ds-lemma} + +\textbf{Example 1 --- GBM, derive explicit solution:} + +SDE: \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\). + +\textbf{Goal:} Find \(S_t\) in closed form. + +\textbf{Step 1} --- Guess \(F(t,x) = \log x\). Compute partials: \(\partial_t F = 0\), \(\partial_x F = 1/x\), \(\partial_{xx}F = -1/x^2\). + +\textbf{Step 2} --- Apply Itō's lemma: + +\[d(\log S_t) = 0 + \frac{1}{S_t}\,dS_t + \tfrac12\cdot(-\tfrac{1}{S_t^2})\cdot\sigma^2 S_t^2\,dt\] + +\[= \frac{\mu S_t\,dt + \sigma S_t\,dW_t}{S_t} - \tfrac12\sigma^2\,dt\] + +\[= \left(\mu - \tfrac12\sigma^2\right)\,dt + \sigma\,dW_t.\] + +\textbf{Step 3} --- Integrate (deterministic integral + Itō integral): + +\[\log S_T - \log S_0 = \left(\mu-\tfrac12\sigma^2\right)T + \sigma W_T.\] + +\textbf{Step 4} --- Exponentiate: + +\[\boxed{S_T = S_0\exp\!\left[\left(\mu - \tfrac12\sigma^2\right)T + \sigma W_T\right].}\] + +The \textbf{Itō correction} \(-\tfrac12\sigma^2 T\) lowers the expected log-return: \(\mathbb{E}[\log S_T] = \log S_0 + (\mu-\tfrac12\sigma^2)T\), but \(\mathbb{E}[S_T] = S_0 e^{\mu T}\) (Jensen's inequality explains the gap: \(e^{\mathbb{E}[X]} < \mathbb{E}[e^X]\) for non-degenerate \(X\)). + +\begin{verbatim} +log S_t +| Ito correction: slope = mu - sigma^2/2 (lower than mu!) +| _.-' +| / <- Without Ito correction, slope would be mu (wrong!) +|/ ++----------------------------> t + +E[log S_t] = log S_0 + (mu - sigma^2/2)*t + (time-averaged growth, always less than mu for sigma > 0) +\end{verbatim} + +\textbf{Example 2 --- Itō product rule (\(d(X_t Y_t)\)):} + +By Itō's lemma applied to \(F(x,y) = xy\): + +\[d(X_t Y_t) = Y_t\,dX_t + X_t\,dY_t + d[X,Y]_t\] + +where \(d[X,Y]_t = \sigma_X\sigma_Y\,dt\). Compare to ordinary calculus: \(d(xy) = y\,dx + x\,dy\) (no cross term because \((dx)^2=0\)). + +\textbf{Example 3 --- Integration by parts for stochastic integrals:} + +\[\int_0^T W_t\,dW_t = \tfrac12 W_T^2 - \tfrac12 T.\] + +Ordinary calculus would give \(\int_0^T W_t\,dW_t = \tfrac12 W_T^2\). The \(-\tfrac12 T\) correction comes from the quadratic variation. + +\textbf{Verification via Itō's lemma:} Set \(F(t,x) = x^2/2\): \(dF = x\,dW + \tfrac12\cdot 1 \cdot dt = W_t\,dW_t + \tfrac12\,dt\). Integrate: \(\tfrac12 W_T^2 - 0 = \int_0^T W_t\,dW_t + \tfrac12 T\) → result follows. ✓ + +\paragraph{2.2.3 Itō vs Stratonovich}\label{itux14d-vs-stratonovich} + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2222}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2889}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.4889}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Property +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Itō integral +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Stratonovich integral (\(\circ\)) +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Chain rule & Modified (\(+\tfrac12\sigma^2\partial_{xx}F\) term) & Standard calculus chain rule \\ +Martingale & Yes (if \(f\) adapted) & No in general \\ +Use in finance & Natural (no look-ahead) & Physics, geometry \\ +Conversion & \(\int f\circ dW = \int f\,dW + \tfrac12\int \partial_x f\,\sigma\,dt\) & (same identity) \\ +SDE solutions & Different numerics needed & Standard ODE methods work \\ +\end{longtable} +} + +\textbf{Conversion formula} --- Itō \(\to\) Stratonovich: + +\[\int_0^T f(X_t)\circ dW_t = \int_0^T f(X_t)\,dW_t + \tfrac{1}{2}\int_0^T f'(X_t)\sigma_t\,dt.\] + +\textbf{Rule of thumb:} Use Itō in finance (causality, no-arbitrage); use Stratonovich in physics/differential geometry (coordinate-invariant chain rule). + +\subsubsection{2.3 General Itō SDEs}\label{general-itux14d-sdes} + +\[dX_t = b(t, X_t)\,dt + \boldsymbol{\sigma}(t, X_t)\,dW_t,\quad X_0 = x_0.\] + +\paragraph{2.3.0 Existence, Uniqueness and Picard Iteration}\label{existence-uniqueness-and-picard-iteration} + +\begin{tcolorbox}[colback=thmbg!60,colframe=optgold,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Theorem / Key Idea}} — Theorem — Strong Solution Existence (Picard–Lindelöf for SDEs)},breakable] +If $b$ and $\sigma$ are **globally Lipschitz** in $x$ (uniformly in $t$): +$\|b(t,x)-b(t,y)\| + \|\sigma(t,x)-\sigma(t,y)\| \le L\|x-y\|$, + +and satisfy **linear growth**: $\|b(t,x)\|^2 + \|\sigma(t,x)\|^2 \le C^2(1+\|x\|^2)$, + +then there exists a **unique strong solution** with $\mathbb{E}\!\left[\sup_{t\le T}\|X_t\|^2\right] < \infty$. +\end{tcolorbox} + +\textbf{Picard iteration --- construct the solution step by step:} + +Set \(X_t^{(0)} = x_0\) (constant). For \(n\ge 0\): + +\[X_t^{(n+1)} = x_0 + \int_0^t b(s, X_s^{(n)})\,ds + \int_0^t \sigma(s, X_s^{(n)})\,dW_s.\] + +\textbf{Intermediate step --- bound the error:} + +Let \(\varepsilon_n(t) = \mathbb{E}\!\left[\sup_{s\le t}|X_s^{(n+1)}-X_s^{(n)}|^2\right]\). + +By Doob's \(L^2\)-inequality and Lipschitz: + +\[\varepsilon_{n+1}(t) \le 2(L^2 T + L^2)\int_0^t \varepsilon_n(s)\,ds.\] + +By induction: \(\varepsilon_n(t) \le C \cdot \frac{(2L^2(T+1)t)^n}{n!} \to 0\). Geometric series → \(X^{(n)}\) is Cauchy in \(L^2\) → converges to the unique solution. + +\textbf{Intuition:} + +\begin{verbatim} +Picard iteration (simplest case: dx = f(x) dt): + +X^(0): --------- x_0 (constant) +X^(1): x_0 + int_0^t f(x_0) ds = x_0 + f(x_0)*t (linear approx) +X^(2): x_0 + int_0^t f(X^(1)) ds (use X^(1) in drift) +X^(3): ... + +Each iteration captures one more "correction layer": + n=0: constant n=1: linear n=2: quadratic ... +\end{verbatim} + +\paragraph{2.3.1 The Fokker-Planck Equation --- How Densities Evolve}\label{the-fokker-planck-equation-how-densities-evolve} + +If \(X_t\) has density \(p(t,x)\), then \(p\) satisfies the \textbf{Fokker-Planck (Kolmogorov forward) PDE}: + +\[\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[b(t,x)\,p] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(t,x)\,p].\] + +\textbf{Derivation sketch:} For any test function \(\phi\): + +\[\frac{d}{dt}\mathbb{E}[\phi(X_t)] = \mathbb{E}[\mathcal{L}\phi(X_t)] = \mathbb{E}\!\left[b\,\phi' + \tfrac12\sigma^2\phi''\right]\] + +using Itō's lemma on \(\phi(X_t)\). Integration by parts in the \(x\)-integral transfers derivatives from \(\phi\) to \(p\), giving the Fokker-Planck equation. + +\textbf{Visual --- density flows rightward (positive drift) and spreads (positive diffusion):} + +\begin{verbatim} +p(t, x) — probability density at time t + +t=0: | peak at x_0 + | *** + | * * <- narrow initial density + | * * + +-----------------> x + x_0 + +t=T/2: | peak shifted right + | *** + | * * <- wider (diffusion spreads it) + | * * + +-----------------> x + x_0 + mu*T/2 + +t=T: | peak drifted further + | *** + | * * <- even wider + | * * + +--------------------------> x + x_0 + mu*T +\end{verbatim} + +\textbf{For OU: \(b = \kappa(\theta-x)\), \(\sigma\) = const} → stationary solution \(p_\infty(x) = \mathcal{N}(\theta, \sigma^2/2\kappa)\). + +\paragraph{2.3.2 Common SDE Reference Table}\label{common-sde-reference-table} + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.1304}} + >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.0725}} + >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2754}} + >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2464}} + >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2754}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Process +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +SDE +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Closed-form \(X_t\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Stationary dist. +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Use in Optimiz-rs +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Brownian motion & \(dX = \sigma\,dW\) & \(X_0 + \sigma W_t\) & --- & Noise baseline \\ +Geometric BM & \(dX = \mu X\,dt + \sigma X\,dW\) & \(X_0 e^{(\mu-\sigma^2/2)t+\sigma W_t}\) & Log-normal & Price model \\ +Ornstein-Uhlenbeck & \(dX = \kappa(\theta-X)\,dt + \sigma\,dW\) & (see §2.4) & \(\mathcal{N}(\theta, \sigma^2/2\kappa)\) & Spread model \\ +CIR & \(dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW\) & (Bessel process) & Gamma\((2\kappa\theta/\sigma^2, \sigma^2/2\kappa)\) & Volatility, rates \\ +SABR & \(dF = \sigma F^\beta dW^1\), \(d\sigma = \nu\sigma\,dW^2\) & (no closed form) & --- & Volatility model \\ +\end{longtable} +} + +\paragraph{2.3.3 Numerical Schemes for SDEs}\label{numerical-schemes-for-sdes} + +When no closed form exists, discretize with step \(\Delta t\): + +\textbf{Euler-Maruyama} (simplest, strong order 0.5): + +\[X_{t+\Delta t} \approx X_t + b(t,X_t)\,\Delta t + \sigma(t,X_t)\,\Delta W_t\] + +where \(\Delta W_t = \sqrt{\Delta t}\,Z\), \(Z\sim\mathcal{N}(0,1)\). + +\textbf{Milstein} (includes first-order Itō correction, strong order 1.0): + +\[X_{t+\Delta t} \approx X_t + b\,\Delta t + \sigma\,\Delta W_t + \tfrac12\sigma\,\sigma_x\bigl[(\Delta W_t)^2 - \Delta t\bigr].\] + +The extra term \(\tfrac12\sigma\sigma_x[(\Delta W_t)^2 - \Delta t]\) comes from applying Itō's lemma to \(\sigma(X_t)dW_t\). + +\begin{verbatim} +Error comparison (log scale): + + | Euler-Maruyama (slope = 0.5) + | * + | * + | * + | * Milstein (slope = 1.0) + | o + | o + | o + +----------------------------> log(Delta_t) + (Milstein converges MUCH faster: halving dt reduces error by 4x instead of 2x) +\end{verbatim} + +\subsubsection{2.4 Ornstein-Uhlenbeck (Mean-Reversion)}\label{ornstein-uhlenbeck-mean-reversion} Used in Optimiz-rs's \texttt{sparse\_mean\_reversion} and \texttt{ou\_estimator} modules: -\begin{equation} - dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t. -\end{equation} -\textbf{Closed-form solution:} -\begin{equation} - X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s. -\end{equation} +\[dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t.\] -\textbf{Half-life:} $\tau_{1/2} = \ln 2/\kappa$. With $\kappa=0.2$/day, half-life $\approx 3.5$ days. +\textbf{Intuition --- restoring force:} The drift is a spring pulling \(X_t\) back to \(\theta\): -\textbf{MLE log-likelihood} (discrete observations at spacing $\Delta t$): -\begin{equation} - \ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[ - \log(2\pi\hat\sigma_i^2) + \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma_i^2} - \right], -\end{equation} -where -$\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}$ and -$\hat\sigma_i^2 = \dfrac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})$. +\begin{verbatim} +X_t + | upper band: theta + sigma/sqrt(2*kappa) + | .--. . ___ <-- random excursions + | / v /\ / \ + |/ v / \ ----- long-run mean: theta (equilibrium) + | \ \.... + | \___/ + | lower band: theta - sigma/sqrt(2*kappa) + +-------------------------------> t -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Jump Processes} -%% ═══════════════════════════════════════════════════════════════════════════ + Arrows: mean-reversion pull toward theta with speed kappa. + Half-life = ln(2) / kappa. + + Strong kappa: tight oscillations around theta (faster spring) + Weak kappa: slow drift back (loose spring, more "random walk" like) +\end{verbatim} + +\paragraph{2.4.1 Closed-Form Solution --- Step by Step}\label{closed-form-solution-step-by-step} + +\textbf{Step 1 --- Integrating factor.} Rewrite the SDE as: + +\[dX_t + \kappa X_t\,dt = \kappa\theta\,dt + \sigma\,dW_t.\] + +Multiply both sides by the integrating factor \(e^{\kappa t}\) and recognize the left-hand side: + +\[d\!\left(e^{\kappa t}X_t\right) = e^{\kappa t}dX_t + \kappa e^{\kappa t}X_t\,dt = e^{\kappa t}\kappa\theta\,dt + e^{\kappa t}\sigma\,dW_t.\] + +(Here we used Itō's product rule: \(d(e^{\kappa t}X_t) = e^{\kappa t}dX_t + X_t\cdot\kappa e^{\kappa t}dt\) --- no quadratic variation cross term since \(e^{\kappa t}\) is deterministic.) + +\textbf{Step 2 --- Integrate both sides from \(0\) to \(t\):} + +\[e^{\kappa t}X_t - X_0 = \kappa\theta\int_0^t e^{\kappa s}\,ds + \sigma\int_0^t e^{\kappa s}\,dW_s\] + +\[e^{\kappa t}X_t - X_0 = \theta(e^{\kappa t} - 1) + \sigma\int_0^t e^{\kappa s}\,dW_s.\] + +\textbf{Step 3 --- Divide by \(e^{\kappa t}\):} + +\[\boxed{X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.}\] + +\textbf{Interpretation of each term:} + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.4000}} + >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.6000}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Term +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Meaning +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +\(\theta\) & Long-run equilibrium (the ``anchor'') \\ +\((X_0-\theta)e^{-\kappa t}\) & Deterministic decay: initial displacement shrinks at rate \(\kappa\) \\ +\(\sigma\int_0^t e^{-\kappa(t-s)}dW_s\) & Stochastic part: weighted sum of all past noise shocks, with \textbf{exponential forgetting} \\ +\end{longtable} +} + +The stochastic integral \(I_t = \sigma\int_0^t e^{-\kappa(t-s)}dW_s\) is a \textbf{Gaussian} random variable (linear functional of Brownian motion) with: + +\[\mathbb{E}[I_t] = 0, \qquad \operatorname{Var}(I_t) = \sigma^2\int_0^t e^{-2\kappa(t-s)}\,ds = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t}).\] + +\textbf{Step 4 --- Marginal distribution:} + +\[X_t \sim \mathcal{N}\!\left(\theta + (X_0-\theta)e^{-\kappa t},\;\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})\right).\] + +As \(t\to\infty\): \(X_t \to \mathcal{N}(\theta, \sigma^2/2\kappa)\) --- the stationary distribution. + +\paragraph{\texorpdfstring{2.4.2 Transition Density (Conditional on \(X_s\))}{2.4.2 Transition Density (Conditional on X\_s)}}\label{transition-density-conditional-on-x_s} + +\[X_t \mid X_s \sim \mathcal{N}\!\left(\theta + (X_s-\theta)e^{-\kappa(t-s)},\;\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa(t-s)})\right), \quad t > s.\] + +This is exact (no approximation) because the OU process is \textbf{linear}. Key formulas: + +\[\hat\mu(\tau) = \theta + (X_s-\theta)e^{-\kappa\tau}, \qquad \hat\sigma^2(\tau) = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\tau}), \quad \tau=t-s.\] + +\begin{verbatim} +Transition density spreading over time: + + p(x_t | x_0 = x_0) + +t=0: delta function at x_0 + | + |* <- spike + +t=0.2: | **** + |* * <- bell curve, mean shifted toward theta + +t=T: | **** + | * * <- centered at theta, wider + | * * + +-----------> x +\end{verbatim} + +\paragraph{2.4.3 Half-Life and Mean-Reversion Speed}\label{half-life-and-mean-reversion-speed} + +\textbf{Half-life:} \(\tau_{1/2} = \ln 2/\kappa\) --- time for the initial displacement to halve. + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{}lll@{}} +\toprule\noalign{} +\(\kappa\) (per year) & Half-life & Typical use \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +0.2 & 3.5 yr & Long-term macro factors \\ +10 & 25 days & Cross-sectional equity spreads \\ +55 & 4.6 days & Short-term pair spreads \\ +252 & 1 trading day & Intraday alpha signals \\ +\end{longtable} +} + +\textbf{MLE log-likelihood} (discrete observations at spacing \(\Delta t\)): + +\[\ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[\log(2\pi\hat\sigma^2) + \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma^2}\right],\] + +where \(\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}\) and \(\hat\sigma^2 = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})\). + +\textbf{Score equations} (differentiate \(\ell\) and set to zero): + +\[\frac{\partial\ell}{\partial\theta} = \sum_i \frac{X_{t_i}-\hat\mu_i}{\hat\sigma^2}(1-e^{-\kappa\Delta t}) = 0,\] + +\[\frac{\partial\ell}{\partial\kappa} = \sum_i \frac{(X_{t_i}-\hat\mu_i)}{\hat\sigma^2}(X_{t_{i-1}}-\theta)\Delta t\,e^{-\kappa\Delta t} - \sum_i \frac{\partial\log\hat\sigma^2}{\partial\kappa} = 0.\] + +These are nonlinear in \(\kappa\); Optimiz-rs solves them with DE (\texttt{ou\_estimator::fit\_mle()}). + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Calibrating OU to an Equity-Pair Spread},breakable] +**Data:** Daily log-spread $X_t = \log(P_A / P_B)$ for a co-integrated pair, +$n=250$ observations, $\Delta t=1/252$ years. + +**Step 1 — MLE:** Maximize $\ell(\kappa, \theta, \sigma)$ using `ou_estimator::fit_mle()`. + +**Step 2 — Intermediate verification:** The OU log-likelihood surface: + +``` +ell(kappa, theta | sigma_hat) + +kappa + ^ + | +++ <- log-lik thickening near true kappa + | +++++ + | +++++++ + | +++++ + | +++ + | + +-------------------------> theta + (theta_hat is the sample mean of X_t, very well identified) + (kappa is harder: need long series to identify mean-reversion speed) +``` + +**Typical results:** + +| Parameter | Estimate | Interpretation | +|-----------|----------|----------------| +| $\hat\kappa$ | 55/yr | half-life approx 4.6 days | +| $\hat\theta$ | 0.003 | long-run spread approx 0.3% | +| $\hat\sigma$ | 0.12/yr$^{0.5}$ | daily spread vol approx 0.75% | + +**Step 3 — Diagnostic:** + +Standardized residuals: $r_i = (X_{t_i} - \hat\mu_i)/\hat\sigma$ should be $\mathcal{N}(0,1)$. + +``` +r_i histogram vs N(0,1) + + | . + | . . <- histogram bars (sample) + |. . ......... + | .......... <- N(0,1) curve (theory) + | . . + +------|------|-----> r_i + -2 +2 +``` + +Ljung-Box test: checks for remaining autocorrelation in $r_i$. + +**Step 4 — Trading signal:** +Enter when $|X_t - \hat\theta| > 2\hat\sigma_\infty$ where $\hat\sigma_\infty = \hat\sigma/\sqrt{2\hat\kappa}$. +Exit at $X_t = \hat\theta$. Expected holding time $\approx \hat\tau_{1/2} = \ln 2/\hat\kappa \approx 4.6$ days. + +**P&L decomposition:** +- Gross expected profit per trade $\approx 2\hat\sigma_\infty = 2\hat\sigma/\sqrt{2\hat\kappa}$. +- Transaction costs must be $< 2\hat\sigma_\infty$ for profitability. +\end{tcolorbox} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{3 · Jump Processes}\label{jump-processes} Many financial time series exhibit sudden large moves that Brownian motion cannot capture. -\subsection{Poisson Process} +\subsubsection{3.1 Poisson Process}\label{poisson-process} -\begin{defbox}{Poisson Process} -A counting process $N = (N_t)_{t\ge 0}$ is a \emph{Poisson process with intensity} -$\lambda > 0$ if: -\begin{enumerate} - \item $N_0 = 0$. - \item Independent, stationary increments. - \item $\PP(N_{t+h}-N_t=1) = \lambda h + o(h)$ and $\PP(\Delta N > 1) = o(h)$. -\end{enumerate} -Equivalently, $N_t \sim \text{Poisson}(\lambda t)$ and inter-arrival times are $\text{Exp}(\lambda)$. -The \emph{compensated} process $\tilde{N}_t = N_t - \lambda t$ is a martingale. -\end{defbox} +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Poisson Process},breakable] +A counting process $N = (N_t)_{t\ge 0}$ is a *Poisson process with +intensity* $\lambda > 0$ if: -\subsection{Compound Poisson Jump-Diffusion (Merton 1976)} +1. $N_0 = 0$. +2. Independent, stationary increments. +3. $\mathbb{P}(N_{t+h}-N_t=1) = \lambda h + o(h)$ and $\mathbb{P}(\Delta N > 1) = o(h)$. +\end{tcolorbox} -\begin{equation} - \frac{dS_t}{S_{t^-}} = \mu\,dt + \sigma\,dW_t - + d\!\left(\sum_{k=1}^{N_t}(e^{J_k}-1)\right), -\end{equation} -with $N_t \sim \text{Poisson}(\lambda)$ and $J_k \sim \mathcal{N}(\mu_J, \sigma_J^2)$. +Equivalently, \(N_t \sim \text{Poisson}(\lambda t)\) and inter-arrival times are \(\text{Exp}(\lambda)\). The \emph{compensated} process \(\tilde N_t = N_t - \lambda t\) is a martingale. -\textbf{Merton option price} — a Poisson mixture of Black--Scholes prices: -\begin{equation} - C_{\text{Merton}} = \sum_{n=0}^\infty \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} - \cdot C_{\text{BS}}\!\left(S_0, K, T, r_n, \sigma_n^2\right), -\end{equation} -where -\begin{align*} - \lambda' &= \lambda\, e^{\mu_J+\frac12\sigma_J^2},\\ - r_n &= r - \lambda\bigl(e^{\mu_J+\frac12\sigma_J^2}-1\bigr) + \frac{n(\mu_J+\tfrac12\sigma_J^2)}{T},\\ - \sigma_n^2 &= \sigma^2 + \frac{n\sigma_J^2}{T}. -\end{align*} +\textbf{Sample path --- step function with random jumps (\(\lambda=2\) per unit time):} -\subsection{Lévy Processes and the Lévy--Khintchine Representation} +\begin{verbatim} +N_t + 5 | ___________ + 4 | ___________ + 3 | ______ + 2 | ______ + 1 |____ + 0 | + +-----|------|-------|------|---------> t + tau1 tau2 tau3 tau4 + (Exp(lambda) inter-arrivals, each tau_i ~ Exp(2)) +\end{verbatim} -\begin{thmbox}{Lévy--Khintchine} -Every Lévy process (independent, stationary increments) has characteristic function -\begin{equation} - \E[e^{i\xi X_t}] = \exp\!\left(t\left[ - ib\xi - \tfrac{1}{2}\sigma^2\xi^2 - + \int_{\R\setminus\{0\}} \bigl(e^{i\xi z}-1-i\xi z\,\mathbf{1}_{|z|\le1}\bigr)\nu(dz) - \right]\right), -\end{equation} -where $(b, \sigma^2, \nu)$ is the \emph{Lévy triplet} and $\nu$ the \emph{Lévy measure}, +\subsubsection{3.2 Compound Poisson Jump-Diffusion (Merton 1976)}\label{compound-poisson-jump-diffusion-merton-1976} + +\[\frac{dS_t}{S_{t^-}} = \mu\,dt + \sigma\,dW_t + d\Bigl(\sum_{k=1}^{N_t}(e^{J_k}-1)\Bigr),\] + +with \(N_t\) Poisson(\(\lambda\)) and \(J_k \sim \mathcal{N}(\mu_J, \sigma_J^2)\). + +\textbf{Sample path --- smooth diffusion interrupted by sudden jumps:} + +\begin{verbatim} +S_t +| ^ jump +15% +| /| +| / | +| / | v jump -20% +| / \\ /| +| / \\_______/ | +| / \\____ +| / \\... +| / ++---------------------------------> t + (Brownian between jumps; jump times ~ Poisson) +\end{verbatim} + +\textbf{Merton option price} --- Poisson mixture of Black-Scholes prices: + +\[C_{\text{Merton}} = \sum_{n=0}^\infty \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} +\cdot C_{\text{BS}}\!\left(S_0, K, T, r_n, \sigma_n^2\right),\] + +where \(\lambda' = \lambda e^{\mu_J+\frac12\sigma_J^2}\), \(r_n = r - \lambda(e^{\mu_J+\frac12\sigma_J^2}-1) + n(\mu_J+\tfrac12\sigma_J^2)/T\), and \(\sigma_n^2 = \sigma^2 + n\sigma_J^2/T\). + +\textbf{Intuition:} Condition on exactly \(n\) jumps occurring (probability \(e^{-\lambda' T}(\lambda' T)^n/n!\)). In that scenario the world is a BS world with adjusted drift \(r_n\) and total variance \(\sigma^2 T + n\sigma_J^2\). Average over the Poisson distribution of \(n\). + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Fitting Merton to a Crash Event},breakable] +**Observed:** S&P 500, March 2020. Implied vol surface shows a vol smile — +OTM puts are expensive (fat left tail), which pure BS cannot explain. + +**Merton calibration** (4 parameters: $\sigma, \lambda, \mu_J, \sigma_J$): + +| Parameter | Estimated value | Interpretation | +|-----------|----------------|----------------| +| $\sigma$ | 0.18/yr | baseline diffusion vol | +| $\lambda$ | 3/yr | approx 3 crash events per year | +| $\mu_J$ | -0.12 | average log-jump = -12% | +| $\sigma_J$ | 0.08 | jump size std = 8% | + +**Fitting procedure:** + +1. Collect implied vols for strikes $K$ and maturities $T$. +2. Minimise $\sum_{K,T}(C_{\text{Merton}}(K,T;\theta) - C_{\text{market}})^2$ + via `differential_evolution` (DE is ideal — 4 params, non-convex landscape). +3. **Diagnostic:** Plot Merton vs market smile; expect fit within 0.5 vega. + +**Result:** Negative $\mu_J$ captures left-tail skew, explaining costly OTM puts. +\end{tcolorbox} + +\subsubsection{3.3 Levy Processes and the Levy-Khintchine Representation}\label{levy-processes-and-the-levy-khintchine-representation} + +\begin{tcolorbox}[colback=thmbg!60,colframe=optgold,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Theorem / Key Idea}} — Theorem — Levy-Khintchine},breakable] +Every Levy process (independent stationary increments) has characteristic function + +$$\mathbb{E}[e^{i\xi X_t}] = \exp\!\Bigl(t\Bigl[i b\xi - \tfrac{1}{2}\sigma^2\xi^2 ++ \int_{\mathbb{R}\setminus\{0\}} \bigl(e^{i\xi z}-1-i\xi z\mathbf{1}_{|z|\le1}\bigr)\nu(dz)\Bigr]\Bigr)$$ + +where $(b, \sigma^2, \nu)$ is the *Levy triplet* and $\nu$ the *Levy measure*, satisfying $\int(1\wedge z^2)\nu(dz)<\infty$. -\end{thmbox} +\end{tcolorbox} -\begin{center} -\begin{tabular}{L{3.5cm} L{5.5cm} L{4cm}} -\toprule -\textbf{Process} & \textbf{Lévy measure $\nu$} & \textbf{Use case} \\ -\midrule -Brownian motion & $\nu=0$ & Continuous diffusion \\[3pt] -Compound Poisson & Finite measure & Rare large jumps \\[3pt] -Variance Gamma & $\nu(dz)\propto e^{-c|z|}/|z|$ & Equity returns \\[3pt] -CGMY & $e^{-G|z|}/|z|^{1+Y}$ (neg), $e^{-Mx}/x^{1+Y}$ (pos) & Heavy tails, $Y\in(0,2)$ \\[3pt] -$\alpha$-stable & $c|z|^{-1-\alpha}$ & Infinite-variance regimes \\ -\bottomrule -\end{tabular} -\end{center} +\textbf{Levy measure tail shapes:} -\subsection{SDEs with Jumps — Generator and Itô Formula} +\begin{verbatim} +nu(dz)/dz -\begin{equation} - dX_t = b(X_{t^-})\,dt + \sigma(X_{t^-})\,dW_t - + \int_{\R} c(X_{t^-}, z)\,\widetilde{N}(dt, dz), -\end{equation} -where $\widetilde{N}(dt,dz) = N(dt,dz) - \nu(dz)\,dt$ is the \emph{compensated jump measure}. +Gaussian BM: nu = 0 (no jump component, only diffusion) + +Compound Poisson (rare large jumps): + | ^ ^ (discrete point masses at fixed jump sizes) + +--*----*---> z + +Variance Gamma (smooth exponential decay): + |\\ + | \\. + | \\...___ + +---------> z (heavier left tail than Gaussian) + +alpha-stable (power-law heavy tail): + |\\ + | \\..... + +-----------> z (infinite variance, very fat tail) +\end{verbatim} + +\textbf{Levy Process Zoo} + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{}lll@{}} +\toprule\noalign{} +Process & Levy measure \(\nu\) & Use case \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Brownian motion & \(\nu=0\) & continuous diffusion \\ +Compound Poisson & finite measure & rare large jumps \\ +Variance Gamma & \(\nu(dz)\propto e^{-c|z|}/|z|\) & equity returns \\ +CGMY & power-law with exponential cutoff & heavy tails, \(Y\in(0,2)\) \\ +\(\alpha\)-stable & \(c|z|^{-1-\alpha}\) & infinite-variance regimes \\ +\end{longtable} +} + +\subsubsection{3.4 SDEs with Jumps --- Generator and Ito Formula}\label{sdes-with-jumps-generator-and-ito-formula} + +\[dX_t = b(X_{t^-})\,dt + \sigma(X_{t^-})\,dW_t ++ \int_{\mathbb{R}} c(X_{t^-}, z)\,\tilde N(dt, dz),\] + +where \(\tilde N(dt,dz) = N(dt,dz) - \nu(dz)\,dt\) is the \emph{compensated jump measure}. + +\textbf{Ito formula for jump-diffusions:} + +\[dF(X_t) = \mathcal{L}F\,dt + \partial_x F\,\sigma\,dW_t ++ \int\bigl[F(X_{t^-}+c)-F(X_{t^-})\bigr]\tilde N(dt,dz),\] -\textbf{Itô formula for jump-diffusions:} -\begin{equation} - dF(X_t) = \mathcal{L}F\,dt + \partial_x F\,\sigma\,dW_t - + \int\bigl[F(X_{t^-}+c)-F(X_{t^-})\bigr]\widetilde{N}(dt,dz), -\end{equation} where the \emph{generator} is -\begin{equation} - \mathcal{L}F = b\,\partial_x F + \tfrac12\sigma^2\partial_{xx}F - + \int\bigl[F(x+c)-F(x)-c\,\partial_x F\bigr]\nu(dz). -\end{equation} -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Optimal Control (HJB, PMP, Jumps)} -%% ═══════════════════════════════════════════════════════════════════════════ +\[\mathcal{L}F = b\,\partial_x F + \tfrac12\sigma^2\partial_{xx}F ++ \int\bigl[F(x+c)-F(x)-c\,\partial_x F\bigr]\nu(dz).\] -\textbf{Big picture.} Optimal control asks: \textit{given a stochastic system we can steer -with a control $u_t$, what policy minimises expected cost?} Three complementary tools -answer this: +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\medskip -\begin{center} -\begin{tabular}{L{2.5cm} L{4cm} L{3cm} L{4cm}} -\toprule -\textbf{Tool} & \textbf{Solves} & \textbf{Scales to} & \textbf{Intuition} \\ -\midrule -HJB PDE & Value function $V(t,x)$ & Low dim (PDE grid) & Dynamic programming \\ -PMP & Optimal paths $(X_t,p_t)$ & High dim (ODE) & Adjoint sensitivity \\ -HJBI & Same as HJB + jumps & Low dim & Non-local integral term \\ -\bottomrule -\end{tabular} -\end{center} +\subsection{4 · Optimal Control (HJB, PMP, Jumps)}\label{optimal-control-hjb-pmp-jumps} -\subsection{Stochastic Hamilton--Jacobi--Bellman Equation} +\textbf{Big picture.} Optimal control asks: \emph{given a stochastic system we can steer with a control \(u_t\), what policy minimises expected cost?} Three complementary tools answer this: + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.1667}} + >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2222}} + >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.3056}} + >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.3056}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Tool +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Solves +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Scales to +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Intuition +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +HJB PDE & Value function \(V(t,x)\) & Low dim (PDE grid) & Dynamic programming \\ +PMP & Optimal paths \((X_t,p_t)\) & High dim (ODE) & Adjoint sensitivity \\ +HJBI & Same as HJB + jumps & Low dim & Non-local integral term \\ +\end{longtable} +} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsubsection{4.1 Stochastic HJB}\label{stochastic-hjb} + +\textbf{Setup.} The state \(X_t \in \mathbb{R}^d\) evolves as + +\[dX_t = b(X_t,u_t)\,dt + \sigma(X_t,u_t)\,dW_t,\] -\textbf{Setup.} The state $X_t \in \R^d$ evolves as -$dX_t = b(X_t,u_t)\,dt + \sigma(X_t,u_t)\,dW_t$, and we minimise the total expected cost -\[ - J(t,x;u) = \E\!\left[\int_t^T \ell(X_s,u_s)\,ds + g(X_T)\,\Big|\,X_t=x\right]. -\] -The \textbf{value function} $V(t,x) = \inf_u J(t,x;u)$ satisfies: -\begin{equation} - \boxed{ - -\partial_t V = \inf_{u\in\mathcal{U}}\Bigl[ - \underbrace{\ell(x,u)}_{\text{running cost}} - + \underbrace{\nabla_x V^{\!\top} b(x,u)}_{\text{1st-order drift}} - + \underbrace{\tfrac12\tr\!\bigl(\sigma\sigma^{\!\top}(x,u)\,\nabla_x^2 V\bigr)}_{\text{curvature correction}} - \Bigr],\quad V(T,\cdot)=g. - } -\end{equation} +\[J(t,x;u) = \mathbb{E}\!\left[\int_t^T \ell(X_s,u_s)\,ds + g(X_T)\,\Big|\,X_t=x\right].\] -Under smooth $V$, the \textbf{feedback law} is -$u^\star(t,x) = \arg\min_u[\ell(x,u)+\nabla_x V^\top b(x,u)]$. +The \textbf{value function} \(V(t,x) = \inf_u J(t,x;u)\) satisfies: -%% ── Example: Merton portfolio ──────────────────────────────────────────── -\begin{tcolorbox}[breakable, enhanced, colback=green!5, colframe=green!40!black, - title={\sffamily\bfseries Example — Optimal Portfolio Allocation (Merton 1969)}, - arc=3pt, boxrule=0.6pt] +\[-\partial_t V = \inf_{u\in\mathcal{U}}\Bigl[\ell(x,u) + \nabla_x V^{\!\top} b(x,u) ++ \tfrac12\operatorname{Tr}\bigl(\sigma\sigma^{\!\top}(x,u)\,\nabla_x^2 V\bigr)\Bigr], +\quad V(T,\cdot)=g.\] + +\textbf{Intuition --- three terms inside the infimum:} - \(\ell(x,u)\) --- instantaneous running cost (pay now). - \(\nabla_x V^\top b\) --- drift of the value function (first-order Taylor in state change). - \(\tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2 V)\) --- curvature correction due to noise (stochastic analogue of the second-order Taylor term). + +Under smooth \(V\), the \textbf{feedback law} is \(u^\star(t,x) = \arg\min_u[\ell(x,u)+\nabla_x V^\top b(x,u)].\) + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Optimal Portfolio Allocation (Merton 1969)},breakable] Investor wealth $X_t$ follows $dX_t = (r + u_t(\mu-r))X_t\,dt + u_t\sigma X_t\,dW_t$, -with $u_t$ the fraction in the risky asset. Maximise $\E[\log X_T]$. +where $u_t\in\mathbb{R}$ is the fraction invested in the risky asset. -\textbf{Ansatz} $V(t,x) = \ln x + f(t)$ substituted into HJB gives: -\[ - f'(t) = -r - \frac{(\mu-r)^2}{2\sigma^2}, \qquad f(T)=0. -\] -\textbf{Optimal Merton rule} (constant, independent of wealth and time): -\begin{equation} - u^\star = \frac{\mu - r}{\sigma^2}. -\end{equation} -Invest a fraction proportional to the Sharpe ratio and inversely to variance. +Minimise $-\mathbb{E}[\log X_T]$ (maximise expected log-utility). + +**Ansatz:** $V(t,x) = \ln x + f(t)$. Substituting into HJB: + +$$f'(t) = -r - \frac{(\mu-r)^2}{2\sigma^2},\qquad f(T)=0.$$ + +The **optimal Merton rule** is constant: + +$$u^\star = \frac{\mu-r}{\sigma^2} \quad (\text{fraction in risky asset}).$$ + +Invest a fixed fraction proportional to the Sharpe ratio, inversely to variance — +independent of wealth and time. \end{tcolorbox} -\textbf{LQR special case} ($\ell = x^\top Q x + u^\top R u$, $b=Ax+Bu$): -$V(t,x)=x^\top P(t)x + v(t)$ with $P$ solving the \emph{matrix Riccati ODE}: -\begin{equation} - -\dot{P} = A^\top P + PA - PBR^{-1}B^\top P + Q,\quad P(T)=Q_T. -\end{equation} -The optimal control is \textbf{linear feedback}: $u^\star_t = -R^{-1}B^\top P(t)X_t$. +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -%% ── Example: Almgren-Chriss ────────────────────────────────────────────── -\begin{tcolorbox}[breakable, enhanced, colback=green!5, colframe=green!40!black, - title={\sffamily\bfseries Example — Optimal Inventory Liquidation (Almgren--Chriss)}, - arc=3pt, boxrule=0.6pt] -Trader liquidates $X_0$ shares by $T$. Inventory $X_t$, rate $u_t < 0$: -\[ - dX_t = u_t\,dt,\qquad \ell(x,u) = \alpha x^2 + \beta u^2. -\] -Deterministic LQR ($\sigma=0$), $A=0$, $B=1$, $Q=\alpha$, $R=\beta$. -Riccati solution gives the optimal TWAP-like schedule: -\begin{equation} - u^\star(t) = -\frac{\kappa}{\sinh(\kappa T)}\cosh(\kappa(T-t))\,X_0, - \qquad \kappa=\sqrt{\alpha/\beta}. -\end{equation} -Large $\kappa$ (high risk aversion, low impact) $\to$ aggressive front-loaded selling. +\textbf{LQR special case} (\(\ell = x^\top Q x + u^\top R u\), \(b=Ax+Bu\), \(\sigma\) constant): \(V(t,x)=x^\top P(t)x + v(t)\) with \(P\) solving the \emph{matrix Riccati ODE}: + +\[-\dot P = A^\top P + PA - PBR^{-1}B^\top P + Q,\quad P(T)=Q_T.\] + +The optimal control is \textbf{linear feedback}: \(u^\star_t = -R^{-1}B^\top P(t)X_t\). + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Optimal Inventory (Almgren-Chriss liquidation)},breakable] +Liquidate $X_0$ shares by time $T$. Inventory $X_t$, trading rate $u_t<0$: + +$$dX_t = u_t\,dt, \quad +\ell(x,u) = \underbrace{\alpha x^2}_{\text{risk}} + \underbrace{\beta u^2}_{\text{impact}}.$$ + +This is a **deterministic LQR** with +$A=0$, $B=1$, $Q=\alpha$, $R=\beta$. +The Riccati solution gives the TWAP-like schedule + +$$u^\star(t,x) = -\frac{\alpha}{\beta}\cdot\frac{\sinh(\kappa(T-t))}{\sinh(\kappa T)}\cdot X_0, +\quad \kappa=\sqrt{\alpha/\beta}.$$ + +Large $\kappa$ (high risk aversion or low impact cost) -> aggressive front-loaded selling. \end{tcolorbox} -\subsection{Pontryagin Maximum Principle (PMP)} +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -The PMP converts the HJB PDE into a \textbf{two-point boundary-value ODE}, -feasible in high dimensions where a grid is intractable. +\subsubsection{4.2 Pontryagin Maximum Principle}\label{pontryagin-maximum-principle} -\begin{thmbox}{Pontryagin Maximum Principle} -Define the \textbf{Hamiltonian} $\mathcal{H}(x,u,p) = \ell(x,u)+p^\top b(x,u)$. -If $(X^\star, u^\star)$ is optimal, there exists a \textbf{costate} (adjoint) process -$p_t$ satisfying: -\begin{equation} - \dot{p}_t = -\nabla_x \mathcal{H}(X_t^\star, u_t^\star, p_t), - \quad p_T = \nabla_x g(X_T^\star), -\end{equation} -with the optimality condition -$u_t^\star = \arg\min_u \mathcal{H}(X_t^\star, u, p_t)$ holding a.e. -\end{thmbox} +The PMP avoids the curse of dimensionality --- it converts HJB into a \textbf{two-point boundary-value ODE} in \((X_t, p_t)\), feasible when a PDE grid is intractable. -\textbf{Costate intuition.} $p_t$ is the \emph{shadow price} of state $X_t$: -\[ - p_t = \nabla_x V(t, X_t^\star) - = \frac{\partial(\text{optimal cost-to-go})}{\partial x}. -\] -Increasing the current state by $dx$ changes future cost by $p_t^\top dx$. -This is the continuous-time analogue of \textbf{backpropagation} through a -dynamical system — the costate = gradient of the loss w.r.t.\ state. +\begin{tcolorbox}[colback=thmbg!60,colframe=optgold,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Theorem / Key Idea}} — Theorem (PMP)},breakable] +Define the **Hamiltonian** $\mathcal{H}(x,u,p) = \ell(x,u)+p^\top b(x,u)$. +If $(X^\star, u^\star)$ is optimal, there exists a **costate** process $p_t$ with: -\textbf{Shooting algorithm:} -\begin{codebox} +$$\dot p_t = -\nabla_x \mathcal{H}(X_t^\star, u_t^\star, p_t),\quad p_T = \nabla_x g(X_T^\star),$$ + +and the optimality condition $u_t^\star = \arg\min_u \mathcal{H}(X_t^\star, u, p_t)$ holds a.e. +\end{tcolorbox} + +\textbf{Costate intuition.} \(p_t\) is the \emph{shadow price} of state \(X_t\): + +\[p_t = \nabla_x V(t, X_t^\star) = \frac{\partial (\text{optimal cost-to-go})}{\partial x}.\] + +This is exactly the adjoint / backpropagation equation of deep learning --- PMP is the continuous-time version of gradient backpropagation through a dynamical system. + +\textbf{Algorithm (shooting method):} + +\begin{verbatim} 1. Guess costate p_0 -2. Integrate forward: dX = b(X, u*(X,p)) dt [state ODE] -3. Integrate backward: dp = -grad_x H(X, u*, p) dt [costate ODE] -4. Check boundary: p_T = grad g(X_T) -5. Not satisfied -> update p_0 (Newton/gradient) -> back to 2 -\end{codebox} +2. Integrate forward: dX = b(X, u*(X,p)) dt (state ODE) +3. Integrate backward: dp = -grad_x H(X, u*, p) dt (costate ODE) +4. Check boundary condition: p_T = grad g(X_T) +5. If not satisfied -> update p_0 (Newton / gradient) -> go to 2 +\end{verbatim} -%% ── Example: PMP for Merton ────────────────────────────────────────────── -\begin{tcolorbox}[breakable, enhanced, colback=green!5, colframe=green!40!black, - title={\sffamily\bfseries Example — PMP Costate for the Merton Problem}, - arc=3pt, boxrule=0.6pt] -With $g(x) = -\!\ln x$, the terminal costate is $p_T = -1/X_T^\star$. -The costate ODE $\dot{p}_t = -p_t(r+u^\star(\mu-r))$ integrates to -\[ - p_t = -\frac{e^{-(T-t)(r+u^\star(\mu-r))}}{X_t^\star}, -\] -confirming $p_t = \partial_x V = 1/x$ (log-utility shadow price is inverse wealth). -Optimality $\partial_u\mathcal{H}=0$ recovers $u^\star = (\mu-r)/\sigma^2$. +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — PMP for the Merton Problem},breakable] +With $\ell = 0$, $g(x) = -\ln x$, $b = (r+u(\mu-r))x$, +the Hamiltonian is $\mathcal{H}(x,u,p) = p(r+u(\mu-r))x$. + +**Costate ODE:** +$\dot p_t = -\partial_x \mathcal{H} = -p_t(r+u^\star(\mu-r))$, +with terminal $p_T = -1/X_T^\star$. + +**Optimality condition** $\partial_u\mathcal{H}=0$ recovers $u^\star = (\mu-r)/\sigma^2$. + +The costate path $p_t = -e^{-(T-t)(r+(\mu-r)u^\star)}/X_t^\star$ confirms that the +shadow price scales inversely with wealth — poorer investors value state more. \end{tcolorbox} -The costate pair $(X_t^\star, p_t)$ moves along Hamiltonian geodesics on -$T^\star\R^d$ — a direct link to symplectic geometry (\S\ref{sec:symplectic}). +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\subsection{HJB with Jumps (HJBI)} +The costate pair \((X_t^\star, p_t)\) moves along Hamiltonian geodesics on \(T^\star\mathbb{R}^d\) --- a direct link to symplectic geometry (§10.4). -When the state can jump (\S3.4), the HJB equation gains a -\textbf{non-local integral operator}: -\begin{equation} - -\partial_t V = \inf_{u}\Bigl[ - \ell + \nabla V^\top b + \tfrac12\tr(\sigma\sigma^\top\nabla^2 V) - + \underbrace{\int\bigl[V(x+c)-V(x)-\nabla V^\top c\bigr]\nu(dz)}_{\text{expected value change from jumps}} - \Bigr]. -\end{equation} +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\textbf{Intuition for the integral term.} A jump of size $c$ moves the state from $x$ -to $x+c$, changing the value function by $V(x+c)-V(x)$. The compensator -$\nabla V^\top c$ subtracts the linear part already counted by the drift, -following Itô's formula for jump processes. +\subsubsection{4.3 HJB with Jumps (HJBI)}\label{hjb-with-jumps-hjbi} -%% ── Example: jumps in inventory ───────────────────────────────────────── -\begin{tcolorbox}[breakable, enhanced, colback=green!5, colframe=green!40!black, - title={\sffamily\bfseries Example — Optimal Execution with Jump Risk}, - arc=3pt, boxrule=0.6pt] +When the state can jump (§3.4), the HJB equation gains a \textbf{non-local integral operator}: + +\[-\partial_t V = \inf_{u}\Bigl[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2 V) ++ \underbrace{\int\bigl[V(x+c(x,u,z))-V(x)-\nabla V^\top c(x,u,z)\bigr]\nu(dz)}_{\text{expected value change from jumps}}\Bigr].\] + +\textbf{Intuition for the integral term.} A jump of size \(c\) moves the state from \(x\) to \(x+c\), changing the value function by \(V(x+c)-V(x)\). The compensator \(\nabla V^\top c\) subtracts the linear part already counted in the drift. + +The \texttt{optimal\_control} module discretises the integral on truncated support using Gaussian quadrature. + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Optimal Execution with Jump Risk},breakable] Extend the inventory model with Poisson order-flow shocks: -\[ - dX_t = u_t\,dt + d\!\left(\sum_{k=1}^{N_t} Z_k\right), - \quad Z_k \sim \mathcal{N}(0,\sigma_J^2),\; N_t\sim\text{Poisson}(\lambda). -\] -The HJBI for $V(t,x)$ becomes: -\[ - -\partial_t V = \inf_u\Bigl[\alpha x^2 + \beta u^2 + \partial_x V\,u - + \lambda\,\E_z[V(x+z)-V(x)-z\,\partial_x V]\Bigr]. -\] -With Gaussian jumps the expectation evaluates to -$\tfrac12\lambda\sigma_J^2\,\partial_{xx}V$, so the HJBI reduces to the same Riccati -ODE but with \textbf{effective diffusion} $\sigma_{\rm eff}^2 = \lambda\sigma_J^2$: -order-flow risk acts like additional Brownian volatility, accelerating the sell schedule. + +$$dX_t = u_t\,dt + \Delta J_t,\quad \Delta J_t \sim \text{Compound Poisson}(\lambda, \mathcal{N}(0,\sigma_J^2)).$$ + +With Gaussian jumps, the HJBI reduces to the same LQR Riccati ODE but with +**effective diffusion** $\sigma_{\text{eff}}^2 = \lambda\sigma_J^2$. + +Key insight: order-flow risk acts like additional Brownian volatility, accelerating +the optimal sell schedule. \end{tcolorbox} -\subsection{Viscosity Solutions} +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -When $V$ fails to be $C^{1,2}$ (degenerate diffusion, constraints, non-smooth terminal -payoff), classical solutions may not exist. \textbf{Viscosity solutions} -(Crandall--Lions 1983) provide a rigorous weak notion restoring existence and uniqueness. +\subsubsection{4.4 Viscosity Solutions}\label{viscosity-solutions} -\textbf{Why they matter in practice:} HJB is solved on a grid where $V$ is only -piecewise smooth. Viscosity theory guarantees the numerical scheme converges to -the true solution. +When \(V\) fails to be \(C^{1,2}\) --- degenerate diffusion, constraints, or non-smooth terminal conditions --- classical solutions may not exist. \textbf{Viscosity solutions} (Crandall-Lions 1983) provide a rigorous weak notion that restores existence and uniqueness. -\begin{defbox}{Viscosity Subsolution} -A continuous $V$ is a viscosity \emph{subsolution} if for every smooth $\phi$ -touching $V$ \textbf{from above} at $(t_0,x_0)$ (i.e.\ $V-\phi$ has a local maximum): -\[ - -\partial_t\phi(t_0,x_0) \le - \inf_u\Bigl[\ell(x_0,u) + \nabla_x\phi^\top b + \tfrac12\tr(\sigma\sigma^\top\nabla^2\phi)\Bigr]. -\] -A \emph{supersolution} reverses the inequality with a test touching from \emph{below}. -The unique viscosity \textbf{solution} is simultaneously both. -\end{defbox} +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Viscosity Subsolution},breakable] +A continuous $V$ is a viscosity *subsolution* if for every smooth $\phi$ +touching $V$ **from above** at $(t_0,x_0)$: -\textbf{Practical interpretation.} Classical calculus requires $V$ to satisfy the PDE -pointwise. Viscosity theory says $V$ satisfies it in an averaged sense via smooth test -functions — valid even at kinks and free boundaries. +$$-\partial_t\phi(t_0,x_0) \le \inf_u\Bigl[\ell(x_0,u) + \nabla_x\phi^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2\phi)\Bigr].$$ -%% ── Example: American option ───────────────────────────────────────────── -\begin{tcolorbox}[breakable, enhanced, colback=green!5, colframe=green!40!black, - title={\sffamily\bfseries Example — American Option as a Viscosity Problem}, - arc=3pt, boxrule=0.6pt] -An American put with payoff $g(x)=(K-x)^+$ satisfies the \textbf{variational inequality}: -\[ - \min\Bigl(-\partial_t V - \mathcal{L}_{\rm BS}V,\; V - (K-x)^+\Bigr) = 0, -\] -where $\mathcal{L}_{\rm BS}V = rx\partial_x V + \tfrac12\sigma^2 x^2\partial_{xx}V - rV$. -\begin{itemize} - \item \textbf{Continuation region} ($V>(K-x)^+$): Black--Scholes PDE holds. - \item \textbf{Exercise region} ($V=(K-x)^+$): option exercised immediately. -\end{itemize} -At the free boundary, $\partial_x V$ is continuous (\emph{smooth-pasting}) but -$\partial_{xx}V$ is not — so $V\in C^1$ but $V\notin C^2$. -Viscosity theory handles this kink without modification. +A *supersolution* reverses the inequality. The unique viscosity **solution** is both. \end{tcolorbox} -The backward DP converges to the viscosity solution under the CFL condition -$\Delta t \le C\,(\Delta x)^2$. +\textbf{Practical interpretation:} Classical: ``\(V\) satisfies the PDE pointwise.'' Viscosity: ``\(V\) satisfies the PDE in an averaged sense --- even at kinks.'' -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Mean Field Games (1D Solver)} -%% ═══════════════════════════════════════════════════════════════════════════ +Optimiz-rs's backward DP converges to the viscosity solution under CFL: \(\Delta t \le C\,(\Delta x)^2\). -MFG couples a \textbf{backward HJB} (individual value) with a -\textbf{forward Fokker--Planck} (population density): +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\begin{align} - \text{HJB (backward):}\quad - &{-\partial_t u - \nu\partial_{xx}u + H(x,\partial_x u, m) = 0,} - &u(T,x)&=g(x),\\ - \text{Fokker--Planck (forward):}\quad - &{\partial_t m - \nu\partial_{xx}m - \partial_x(m\,\partial_p H) = 0,} - &m(0,x)&=m_0(x). -\end{align} +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — American Option as a Viscosity Problem},breakable] +American put payoff $g(x) = (K-x)^+$ gives the **variational inequality**: -\textbf{Coupling:} $H$ depends on $m$ (mean-field interaction), creating a fixed-point problem. +$$\min\Bigl(-\partial_t V - \mathcal{L}_{\text{BS}}V,\; V - (K-x)^+\Bigr) = 0.$$ + +- **Continuation region** ($V > g$): Black-Scholes PDE holds. +- **Exercise region** ($V = g$): option exercised immediately. + +At the free boundary: $\partial_x V$ is continuous (*smooth-pasting*) but $\partial_{xx}V$ is not +— $V$ is $C^1$ but not $C^2$. Viscosity theory handles this kink rigorously. +\end{tcolorbox} + +\textbf{Backward DP grid schema:} + +\begin{verbatim} +t=T [ g(x_1) g(x_2) ... g(x_n) ] terminal condition +t=T-1 [ V^1 V^2 ... V^n ] one backward step + . + . +t=0 [ V_0^1 V_0^2 ... V_0^n ] -> optimal policy u*(x,0) +\end{verbatim} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{5 · Mean Field Games (1D Solver)}\label{mean-field-games-1d-solver} + +MFG couples a \textbf{backward HJB} (individual value) with a \textbf{forward Fokker-Planck} (population density): + +\[\begin{aligned} +\text{HJB (backward): } & +-\partial_t u - \nu\partial_{xx}u + H(x,\partial_x u, m) = 0, & u(T,x)&=g(x),\\ +\text{Fokker-Planck (forward): } & +\partial_t m - \nu\partial_{xx}m - \partial_x(m\,\partial_p H) = 0, & m(0,x)&=m_0(x). +\end{aligned}\] + +\textbf{Coupling:} \(H\) depends on \(m\) (mean-field interaction), creating a fixed-point problem. + +\textbf{Backward-forward information flow:} + +\begin{verbatim} +t = 0 t = T + m_0 (known) g(x) (known) + | | + | Fokker-Planck (forward -->) | + | evolves population density m | + | | + v v + m(t,x) <----- mutually consistent --- u(t,x) + HJB (backward <--) + optimal value function + + Each agent uses u to choose optimal control. + Population density m feeds back into u via H(x, du, m). + Fixed point: m and u are simultaneously consistent (Nash equilibrium). +\end{verbatim} \textbf{Fixed-point algorithm:} -\begin{codebox} + +\begin{verbatim} 1. Initialise m^0 = m_0 (e.g. Gaussian) -2. Solve HJB backward -> u^{k+1} +2. Solve HJB backward -> u^{k+1} 3. Extract optimal drift: alpha*(x,t) = -d_p H(x, d_x u^{k+1}, m^k) 4. Solve Fokker-Planck forward with alpha* -> m^{k+1} 5. Check ||m^{k+1} - m^k||_1 < eps; if not, k++ -> go to 2 -\end{codebox} +\end{verbatim} -\textbf{Convergence:} For monotone coupling (Lasry--Lions 2007), the system has a unique -solution and the fixed-point iteration contracts. Monitor both -$\|m^{k+1}-m^k\|_1$ and $\|u^{k+1}-u^k\|_\infty$. +\textbf{Convergence:} For monotone coupling (Lasry-Lions 2007), the system has a unique solution and the fixed-point iteration contracts. -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Kalman Filtering} -%% ═══════════════════════════════════════════════════════════════════════════ +\textbf{Practical tip:} Monitor both \(\|m^{k+1}-m^k\|_1\) and \(\|u^{k+1}-u^k\|_\infty\); divergence of either signals non-monotone coupling or too large a time step. -\subsection{Linear-Gaussian State Space} +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Optimal Liquidation with Many Agents},breakable] +**Setup:** $N \gg 1$ traders each hold $x_t$ shares and must liquidate by $T$. +Aggregate selling rate $\bar u_t = \int u\,m(t,dx)$ depresses the price. -\begin{equation} - \mathbf{x}_t = F\mathbf{x}_{t-1} + \mathbf{w}_t,\quad - \mathbf{w}_t\sim\mathcal{N}(0,Q);\qquad - \mathbf{y}_t = H\mathbf{x}_t + \mathbf{v}_t,\quad - \mathbf{v}_t\sim\mathcal{N}(0,R). -\end{equation} +**Mean-field Hamiltonian:** + +$$H(x, p, m) = \inf_u \Bigl[\alpha x^2 + \beta u^2 + pu\Bigr] ++ \underbrace{\gamma \bar u(m)}_{\text{aggregate impact}}\,x.$$ + +**Nash equilibrium insight:** Each trader liquidates faster when they believe others sell +slowly (first-mover advantage), but this belief is self-defeating in equilibrium. +The MFG fixed point is **more aggressive** than the single-agent Almgren-Chriss schedule +because each agent accounts for crowd impact. +\end{tcolorbox} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{6 · Kalman Filtering}\label{kalman-filtering} + +\subsubsection{6.1 Linear-Gaussian State Space}\label{linear-gaussian-state-space} + +\[\mathbf{x}_t = F\mathbf{x}_{t-1} + \mathbf{w}_t,\; \mathbf{w}_t\sim\mathcal{N}(0,Q); \qquad +\mathbf{y}_t = H\mathbf{x}_t + \mathbf{v}_t,\; \mathbf{v}_t\sim\mathcal{N}(0,R).\] \textbf{Predict:} -\begin{equation} - \hat{\mathbf{x}}^-_t = F\hat{\mathbf{x}}_{t-1},\qquad - P^-_t = FP_{t-1}F^\top+Q. -\end{equation} + +\[\hat{\mathbf{x}}^-_t = F\hat{\mathbf{x}}_{t-1},\quad P^-_t = FP_{t-1}F^\top+Q.\] \textbf{Update:} -\begin{align} - K_t &= P^-_t H^\top(HP^-_t H^\top + R)^{-1},\\ - \hat{\mathbf{x}}_t &= \hat{\mathbf{x}}^-_t + K_t(\mathbf{y}_t - H\hat{\mathbf{x}}^-_t),\\ - P_t &= (I-K_t H)P^-_t. -\end{align} -$K_t$ is the \emph{Kalman gain} — it interpolates between full prior trust ($K\to0$) -and full observation trust ($K\to H^{-1}$). +\[K_t = P^-_t H^\top(HP^-_t H^\top + R)^{-1},\quad +\hat{\mathbf{x}}_t = \hat{\mathbf{x}}^-_t + K_t(\mathbf{y}_t - H\hat{\mathbf{x}}^-_t),\quad +P_t = (I-K_t H)P^-_t.\] -\subsection{Information-Theoretic View} +\(K_t\) is the \emph{Kalman gain} --- it interpolates between full prior trust (\(K\to0\)) and full observation trust (\(K\to H^{-1}\)). -The Kalman filter computes the exact conditional mean -$\hat{\mathbf{x}}_t = \E[\mathbf{x}_t \mid \mathbf{y}_{1:t}]$ in Gaussian models -and minimises $\KL{p(\mathbf{x}_t|\mathbf{y}_{1:t})}{\mathcal{N}(\hat{\mathbf{x}}_t, P_t)}$ -over all Gaussian approximations. +\textbf{Bayesian update --- uncertainty ellipses shrinking:} -\subsection{Continuous-Time Limit (Kalman--Bucy)} +\begin{verbatim} +Before observation (predict): After observation (update): -For $d\mathbf{X}_t = A\mathbf{X}_t\,dt + B\,d\mathbf{W}_t$, -$d\mathbf{Y}_t = C\mathbf{X}_t\,dt + d\mathbf{V}_t$, the error covariance satisfies -the \emph{Riccati ODE}: -\begin{equation} - \dot{P} = AP + PA^\top + BQB^\top - PC^\top R^{-1}CP,\qquad P(0)=P_0. -\end{equation} + +------------------+ +--------+ + | | | | + | p(x | y_1:t-1) | ----> |p(x|y_t)| + | wide ellipse | | tight | + +------------------+ +--------+ -%% ═══════════════════════════════════════════════════════════════════════════ -\section{MCMC (Metropolis--Hastings and Langevin)} -%% ═══════════════════════════════════════════════════════════════════════════ + Kalman gain K interpolates between: + K -> 0 (huge R, ignore y_t) => x_hat = prior + K -> H^-1 (R=0, trust y_t) => x_hat = H^-1 y_t +\end{verbatim} -\subsection{Metropolis--Hastings} +\textbf{Covariance convergence:} \(P_t \to P_\infty\) (algebraic Riccati solution) exponentially fast when \((F,H)\) is observable. -For target $\pi(x) \propto e^{-U(x)}$ and proposal $q(x'\mid x)$: -\begin{equation} - \alpha(x\to x') = \min\!\left(1, \frac{\pi(x')q(x\mid x')}{\pi(x)q(x'\mid x)}\right). -\end{equation} +\subsubsection{6.2 Information-Theoretic View}\label{information-theoretic-view} -\textbf{Detailed balance} $\pi(x)\alpha(x\to x') = \pi(x')\alpha(x'\to x)$ -ensures $\pi$ is the unique stationary distribution. +The Kalman filter computes the exact conditional mean \(\hat{\mathbf{x}}_t = \mathbb{E}[\mathbf{x}_t \mid \mathbf{y}_{1:t}]\) in Gaussian models and minimises \(D_{\mathrm{KL}}(p(\mathbf{x}_t|\mathbf{y}_{1:t})\,\|\,\mathcal{N}(\hat{\mathbf{x}}_t, P_t))\) over all Gaussian approximations. -\textbf{Optimal scaling:} With Gaussian proposal $q(x'|x)=\mathcal{N}(x,h^2 I_d)$, -the step $h^\star \approx 2.38/\sqrt{d}$ (Roberts--Gelman--Gilks 1997) targets $\approx 23$--$45\%$ -acceptance. +\subsubsection{6.3 Continuous-Time Limit (Kalman-Bucy)}\label{continuous-time-limit-kalman-bucy} -\subsection{Langevin Dynamics (MALA)} +For \(d\mathbf{X}_t = A\mathbf{X}_t\,dt + B\,d\mathbf{W}_t\), \(d\mathbf{Y}_t = C\mathbf{X}_t\,dt + d\mathbf{V}_t\), the error covariance satisfies the \emph{Riccati ODE}: + +\[\dot P = AP + PA^\top + BQB^\top - PC^\top R^{-1}CP,\qquad P(0)=P_0.\] + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Tracking a Noisy AR(1) Signal},breakable] +**Model:** Latent trend $x_t = 0.95 x_{t-1} + w_t$ ($Q=0.01$); +noisy observation $y_t = x_t + v_t$ ($R=1.0$). + +Steady-state: $P_\infty \approx 0.17$, so $K_\infty \approx 0.15$. +Kalman weights the new observation at 15%, prior at 85%. + +``` +P_t +1.0 |* +0.8 | \\ +0.6 | \\ +0.4 | \\___ +0.2 | \\__________________ P_inf ~ 0.17 + +--------------------------------> t + (fast convergence to steady-state uncertainty) +``` + +**Implication:** With $R/Q = 100$ (much noisier obs than process), the filter heavily +smooths observations — useful for noisy financial signals like tick prices. +\end{tcolorbox} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{7 · MCMC (Metropolis-Hastings and Langevin)}\label{mcmc-metropolis-hastings-and-langevin} + +\subsubsection{7.1 Metropolis-Hastings}\label{metropolis-hastings} + +For target \(\pi(x) \propto e^{-U(x)}\) and proposal \(q(x'\mid x)\): + +\[\alpha(x\to x') = \min\!\Bigl(1, \frac{\pi(x')q(x\mid x')}{\pi(x)q(x'\mid x)}\Bigr).\] + +\textbf{Detailed balance} \(\pi(x)\alpha(x\to x') = \pi(x')\alpha(x'\to x)\) ensures \(\pi\) is the unique stationary distribution. + +\textbf{Optimal scaling:} With Gaussian proposal, step \(h^\star \approx 2.38/\sqrt{d}\) (Roberts-Gelman-Gilks 1997) targets \textasciitilde23-45\% acceptance. + +\textbf{Energy landscape and accept/reject:} + +\begin{verbatim} +U(x) (negative log-posterior) + + | * * <- local minima (modes of pi) + | / \\ / \\ + | / \\ / \\ + | / *___* \\ <- saddle between modes + | / \\ + +-------------------------> x + + Proposal x' = x + h * xi: + If U(x') < U(x): always accept (moving downhill) + If U(x') > U(x): accept with prob exp(-(U(x')-U(x))) + Prevents getting trapped in local minima. +\end{verbatim} + +\textbf{Trace plot of a well-mixed chain:} + +\begin{verbatim} +x_t ++2 | . . . . . . <- upper mode + | . . . . . . + 0 |... ... <- transitions between modes + | . . . . . +-2 | . . .. <- lower mode + +-----------------------------> t + (crossing between modes = good mixing) +\end{verbatim} + +\subsubsection{7.2 Langevin Dynamics (MALA)}\label{langevin-dynamics-mala} Metropolis-Adjusted Langevin proposal: -\begin{equation} - x' = x - \tfrac{h^2}{2}\nabla U(x) + h\,\xi, \quad \xi\sim\mathcal{N}(0,I_d), -\end{equation} + +\[x' = x - \tfrac{h^2}{2}\nabla U(x) + h\,\xi, \quad \xi\sim\mathcal{N}(0,I_d),\] + a discretisation of the \emph{overdamped Langevin SDE}: -\begin{equation} - dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t, -\end{equation} -whose stationary distribution is exactly $\pi \propto e^{-U}$ (Fokker--Planck analysis). -MALA converges in $O(d^{1/3})$ steps vs $O(d)$ for RW-MH — a key advantage -for high-dimensional posteriors. +\[dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t,\] -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Hidden Markov Models (HMM)} -%% ═══════════════════════════════════════════════════════════════════════════ +whose stationary distribution is exactly \(\pi \propto e^{-U}\). -\subsection{Model} +MALA converges in \(O(d^{1/3})\) steps vs \(O(d)\) for RW-MH --- key advantage for high-dimensional posteriors. -Latent Markov chain $Z_t \in \{1,\ldots,K\}$ with transition matrix -$A_{ij}=\PP(Z_t=j\mid Z_{t-1}=i)$ generates observations -$Y_t \mid Z_t=k \sim B_k(y)$. +\textbf{MALA vs RW-MH trajectory comparison:} -\subsection{Baum--Welch (EM)} +\begin{verbatim} + RW-MH (random walk): MALA (gradient-guided): + . . . ^ -grad U (towards mode) + . . . . --> / + . . . /. . + (diffusive, slow mixing) (directed, fast mixing) +\end{verbatim} -\textbf{E-step (forward--backward):} -\begin{align} - \alpha_t(k) &= B_k(y_t)\sum_j \alpha_{t-1}(j)A_{jk}, & - \beta_t(k) &= \sum_j A_{kj}B_j(y_{t+1})\beta_{t+1}(j).\\[4pt] - \gamma_t(k) &= \frac{\alpha_t(k)\beta_t(k)}{\sum_j \alpha_t(j)\beta_t(j)}, & - \xi_t(j,k) &= \frac{\alpha_t(j)A_{jk}B_k(y_{t+1})\beta_{t+1}(k)}{\mathcal{L}}. -\end{align} +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Calibrating OU Parameters via MCMC},breakable] +**Goal:** Full Bayesian inference on $(\kappa, \theta, \sigma)$ of an OU process. +**Prior:** $\kappa \sim \text{Gamma}(2,0.1)$, $\theta \sim \mathcal{N}(0,1)$, +$\sigma \sim \text{HalfNormal}(0.5)$. + +**MALA chain** ($d=3$, step $h=0.02$): + +``` +Iteration kappa theta sigma log-post +--------- ----- ----- ----- -------- + 1000 42.1 0.003 0.11 125.3 + 2000 55.3 0.003 0.12 128.7 <- burn-in complete + 3000 58.1 0.003 0.12 129.2 + 50000 54.8 0.003 0.12 128.9 <- stable posterior +``` + +**Marginal posterior** (kappa): 95% CI $[44, 67]$, peak at 55/yr — +wider than the MLE point estimate, reflecting genuine parameter uncertainty. +\end{tcolorbox} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{8 · Hidden Markov Models (HMM)}\label{hidden-markov-models-hmm} + +\subsubsection{8.1 Model}\label{model} + +Latent Markov chain \(Z_t \in \{1,\ldots,K\}\) with transition matrix \(A_{ij}=\mathbb{P}(Z_t=j\mid Z_{t-1}=i)\) generates observations \(Y_t \mid Z_t=k \sim B_k(y)\). + +\textbf{State machine diagram (\(K=3\) regimes):} + +\begin{verbatim} + A_12 A_23 + +------------+ +------------+ + | State 1 |------->| State 2 |-------> State 3 + | (Bull) | | (Neutral) | (Bear/Crash) + +------------+<-------+------------+<-------- + A_21 A_32 + + Each state k emits Y_t ~ B_k(y): + B_1: N(mu=+0.05, sigma=0.12) high return, low vol + B_2: N(mu=+0.00, sigma=0.18) flat, medium vol + B_3: N(mu=-0.08, sigma=0.35) crash regime, high vol + + Transition matrix A: + A = [ 0.97 0.02 0.01 ] (row 1: from State 1) + [ 0.01 0.97 0.02 ] (row 2: from State 2) + [ 0.05 0.05 0.90 ] (row 3: from State 3) + (rows sum to 1; diagonal = regime persistence) +\end{verbatim} + +\subsubsection{8.2 Baum-Welch (EM)}\label{baum-welch-em} + +\textbf{E-step (forward-backward):} + +\[\alpha_t(k) = B_k(y_t)\sum_j \alpha_{t-1}(j)A_{jk}, \qquad +\beta_t(k) = \sum_j A_{kj}B_j(y_{t+1})\beta_{t+1}(j).\] + +\[\gamma_t(k) = \frac{\alpha_t(k)\beta_t(k)}{\sum_j \alpha_t(j)\beta_t(j)}, \qquad +\xi_t(j,k) = \frac{\alpha_t(j)A_{jk}B_k(y_{t+1})\beta_{t+1}(k)}{\mathcal{L}}.\] \textbf{M-step:} -\begin{equation} - \hat{A}_{jk} = \frac{\sum_t \xi_t(j,k)}{\sum_t\gamma_t(j)}, \qquad - \hat{\mu}_k = \frac{\sum_t \gamma_t(k)\,y_t}{\sum_t \gamma_t(k)}. -\end{equation} -\textbf{Information-theoretic view:} Baum--Welch is EM on the complete-data log-likelihood; -each iteration monotonically increases $\mathcal{L}(\theta)$ by Jensen's inequality. +\[\hat A_{jk} = \frac{\sum_t \xi_t(j,k)}{\sum_t\gamma_t(j)}, \qquad +\hat\mu_k = \frac{\sum_t \gamma_t(k)\,y_t}{\sum_t \gamma_t(k)}.\] -\textbf{Viterbi (MAP path):} Replace sum-product with max-product: -$\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)$, runs in $O(TK^2)$. +\textbf{Information-theoretic view:} Baum-Welch is EM on the complete-data log-likelihood; each iteration monotonically increases \(\mathcal{L}(\theta)\) by Jensen's inequality. -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Information Theory} -%% ═══════════════════════════════════════════════════════════════════════════ +\textbf{Viterbi trellis diagram (\(K=3\), \(T=4\)):} -\subsection{Entropy and KL Divergence} +\begin{verbatim} +State t=1 t=2 t=3 t=4 + 1 delta1(1) --> delta2(1) --> delta3(1) --> delta4(1) + \ \\ // + 2 delta1(2) --> delta2(2) --> delta3(2) --> delta4(2) + \ \ \ + 3 delta1(3) --> delta2(3) --> delta3(3) --> delta4(3) -\begin{defbox}{KL Divergence} + delta_t(k) = max_j [ delta_{t-1}(j) * A_jk * B_k(y_t) ] + psi_t(k) = argmax (backtrack pointer -> records best prev state) + + After forward pass: trace back from t=T to t=1 via psi to get MAP sequence + z_1*, z_2*, z_3*, z_4* +\end{verbatim} + +\textbf{Viterbi (MAP path):} \(\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)\), \(O(TK^2)\). + +\textbf{Quality check:} Log-likelihood must be non-decreasing; confusion matrix of Viterbi labels vs ground truth validates regime recovery. + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Equity Regime Detection (S&P 500)},breakable] +**Data:** S&P 500 daily log-returns, 2000-2023, $T=5820$ observations. + +**Fit $K=3$ HMM** using `hmm::fit_baum_welch()` with 20 random restarts. + +**Estimated regime parameters:** + +| Regime | Ann. return | Ann. vol | Avg duration | +|--------|------------|---------|-------------| +| Bull | +18% | 10% | 350 days | +| Neutral | +2% | 17% | 80 days | +| Bear | -40% | 38% | 25 days | + +**Smoothed state probabilities** $\gamma_t(k)$: + +``` +P(Bull) 1.0|XXXXXXXXXX XXXXXXXXXX XXXXX + | XXXXXXXX XXXXXXXX + 0.0 +-------------------------> t (years) + 2000 2003 2008 2020 2023 + ^ ^ ^ + dot-com bust GFC COVID crash +``` + +**Use in Optimiz-rs:** Regime beliefs $\gamma_t$ feed as features into +`differential_evolution` to switch risk-aversion $\alpha$ dynamically. +\end{tcolorbox} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{9 · Information Theory}\label{information-theory} + +\subsubsection{9.1 Entropy and KL Divergence}\label{entropy-and-kl-divergence} + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — KL Divergence},breakable] For densities $p, q$: -\begin{equation} - \KL{p}{q} = \int p(x)\log\frac{p(x)}{q(x)}\,dx \;\ge\; 0, -\end{equation} -with equality iff $p=q$ a.e.\ (Gibbs' inequality). Non-symmetric. -\end{defbox} -\textbf{Connection to model selection:} -AIC $= 2k - 2\ln\hat{\mathcal{L}}$ and BIC $= k\ln n - 2\ln\hat{\mathcal{L}}$ -bound $\KL{p_{\text{true}}}{p_\theta}$. +$$D_{\mathrm{KL}}(p\,\|\,q) = \int p(x)\log\frac{p(x)}{q(x)}\,dx \;\ge\; 0,$$ -\subsection{Fisher Information} +with equality iff $p=q$ a.e. (Gibbs inequality). Non-symmetric. +\end{tcolorbox} -\begin{defbox}{Fisher Information Matrix} +\textbf{KL asymmetry --- a critical practical distinction:} + +\begin{verbatim} +p = N(0,1) (narrow Gaussian) q = N(0,4) (wide Gaussian) + + D_KL(p||q): integrate under p. + p lives mostly in [-2,2] where q is large -> small penalty. + D_KL(p||q) is small. (q "covers" p) + + D_KL(q||p): integrate under q. + q places mass in [-6,6]; in tails p is tiny but q is not -> large penalty. + D_KL(q||p) is large. (p does NOT cover q) + + Rule of thumb: + D_KL(p||q): fitting q to match p (mean-seeking, mode-averaging) + D_KL(q||p): q must cover p (mode-seeking, mode-fitting) +\end{verbatim} + +\textbf{Connection to model selection:} AIC \(= 2k - 2\ln\hat{\mathcal{L}}\) and BIC \(= k\ln n - 2\ln\hat{\mathcal{L}}\) bound \(D_{\mathrm{KL}}(p_{\text{true}}\,\|\,p_\theta)\). + +\subsubsection{9.2 Fisher Information}\label{fisher-information} + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Fisher Information Matrix},breakable] For parametric model $p(x;\theta)$: -\begin{equation} - \mathcal{I}(\theta)_{ij} - = \E_{x\sim p}\!\left[\partial_{\theta_i}\log p\;\partial_{\theta_j}\log p\right] - = -\E\!\left[\partial^2_{\theta_i\theta_j}\log p\right]. -\end{equation} -\end{defbox} -\textbf{Cramér--Rao bound:} Any unbiased estimator $\hat{\theta}$ satisfies -$\operatorname{Cov}(\hat{\theta}) \succeq \mathcal{I}(\theta)^{-1}$. -MLE achieves equality asymptotically. +$$\mathcal{I}(\theta)_{ij} += \mathbb{E}_{x\sim p}\!\left[\partial_{\theta_i}\log p\;\partial_{\theta_j}\log p\right] += -\mathbb{E}\!\left[\partial^2_{\theta_i\theta_j}\log p\right].$$ +\end{tcolorbox} -\textbf{Example — Gaussian emission} $B_k = \mathcal{N}(\mu_k,\sigma_k^2)$: -$\mathcal{I}(\mu_k)=\sigma_k^{-2}$,\quad $\mathcal{I}(\sigma_k^2)=(2\sigma_k^4)^{-1}$. +\textbf{Fisher information as curvature of the log-likelihood:} -\subsection{Mutual Information and Feature Relevance} +\begin{verbatim} +log L(theta | x_obs) -\begin{equation} - I(X;Y) = \KL{p(X,Y)}{p(X)p(Y)} = H(X) - H(X\mid Y) \ge 0. -\end{equation} + | .----. + | ./ \. + | ./ \. + | ./ \. + +-----------------------> theta + theta* -\textbf{mRMR criterion} (minimum redundancy, maximum relevance): -\begin{equation} - \max_{Y_i} \Bigl[I(Y_i;\text{target}) - \frac{1}{|S|}\sum_{Y_j\in S}I(Y_i;Y_j)\Bigr]. -\end{equation} + I(theta*) = -d^2/dtheta^2 log L at the peak -\subsection{Natural Gradient (Preview)} + High I (sharp peak): theta well-identified, low estimation variance + Low I (flat peak): theta hard to identify, high estimation variance -Classical gradient descent ignores the geometry of parameter space. The \emph{natural gradient} -replaces $\nabla_\theta\mathcal{L}$ with $\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}$, -giving a reparametrisation-invariant update — see \S\ref{sec:infogeo} for the full development. + Cramer-Rao: Var(theta_hat) >= 1 / I(theta) for any unbiased estimator +\end{verbatim} -%% ═══════════════════════════════════════════════════════════════════════════ -\section{Differential Geometry} -%% ═══════════════════════════════════════════════════════════════════════════ +\textbf{Cramer-Rao bound:} Any unbiased estimator \(\hat\theta\) satisfies \(\operatorname{Cov}(\hat\theta) \succeq \mathcal{I}(\theta)^{-1}\). MLE achieves equality asymptotically. -\subsection{Riemannian Manifolds} +\textbf{Example:} For \(B_k = \mathcal{N}(\mu_k,\sigma_k^2)\): \(\mathcal{I}(\mu_k)=\sigma_k^{-2}\), \(\mathcal{I}(\sigma_k^2)=(2\sigma_k^4)^{-1}\). Higher emission variance -\textgreater{} smaller Fisher info -\textgreater{} less certain parameter estimates. -\begin{defbox}{Riemannian Manifold} -A \emph{Riemannian manifold} $(M, g)$ is a smooth manifold $M$ with a -\emph{metric tensor} $g_p$: a symmetric, positive-definite bilinear form on each +\subsubsection{9.3 Mutual Information and Feature Relevance}\label{mutual-information-and-feature-relevance} + +\[I(X;Y) = D_{\mathrm{KL}}\bigl(p(X,Y)\,\|\,p(X)p(Y)\bigr) = H(X) - H(X\mid Y) \ge 0.\] + +\textbf{Interpretation:} \(I(X;Y)\) = how much knowing \(Y\) reduces uncertainty about \(X\). \(X \perp Y \Rightarrow I=0\). \(Y\) determines \(X\) fully \(\Rightarrow I = H(X)\). + +\textbf{mRMR criterion} (minimum redundancy, maximum relevance) for the sparse module: + +\[\max_{Y_i} \Bigl[I(Y_i;\text{target}) - \frac{1}{|S|}\sum_{Y_j\in S}I(Y_i;Y_j)\Bigr].\] + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Entropy of HMM Regime Probabilities},breakable] +Define discrete regime distribution at time $t$: + +$$\mathbf{p}_t = (\gamma_t(1), \gamma_t(2), \gamma_t(3)).$$ + +**Regime entropy** $H_t = -\sum_k \gamma_t(k)\log \gamma_t(k) \in [0, \log 3]$: + +| Date | P(Bull) | P(Neutral) | P(Bear) | $H_t$ | Certainty | +|------|---------|-----------|---------|-------|-----------| +| 2019-12 | 0.92 | 0.07 | 0.01 | 0.36 | High (Bull clear) | +| 2020-03 | 0.01 | 0.12 | 0.87 | 0.54 | Medium (Bear likely) | +| 2020-06 | 0.42 | 0.45 | 0.13 | 1.05 | Low (mixed) | + +Max entropy $\log 3 \approx 1.10$ = fully uncertain. +**Trading filter:** Only trade when $H_t < 0.7$ (certain regime). +\end{tcolorbox} + +\subsubsection{9.4 Natural Gradient (Preview)}\label{natural-gradient-preview} + +Classical gradient descent ignores parameter-space geometry. The \emph{natural gradient} replaces \(\nabla_\theta\mathcal{L}\) with \(\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}\), giving a reparametrisation-invariant update --- see §10.2 for the full geometric development. + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{10 · Differential Geometry}\label{differential-geometry} + +\subsubsection{10.1 Riemannian Manifolds}\label{riemannian-manifolds} + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Riemannian Manifold},breakable] +A *Riemannian manifold* $(M, g)$ is a smooth manifold $M$ with a +*metric tensor* $g_p$: a symmetric, positive-definite bilinear form on each tangent space $T_p M$. -\end{defbox} +\end{tcolorbox} -\textbf{Geodesics} (locally shortest paths) satisfy: -\begin{equation} - \ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0, -\end{equation} -where -$\Gamma^k_{ij} = \tfrac12 g^{kl}(\partial_i g_{jl}+\partial_j g_{il}-\partial_l g_{ij})$ -are the \emph{Christoffel symbols} encoding intrinsic curvature. +\textbf{Three canonical curvatures:} -\subsection{Information Geometry and Fisher--Rao Metric} -\label{sec:infogeo} +\begin{verbatim} +K > 0 (sphere S2) K = 0 (flat R2) K < 0 (hyperbolic H2) -The statistical manifold $\mathcal{M} = \{p(\cdot;\theta)\}$ carries the -\textbf{Fisher--Rao metric} $g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}$. + (N) | / + /|\ | / + / | \ geodesics --+-- parallel / geodesics +/ | \ reconverge (N-S) | lines / diverge exponentially +\end{verbatim} -\textbf{Natural gradient (Amari 1998):} Steepest descent on $(\mathcal{M}, g)$: -\begin{equation} - \theta \leftarrow \theta - \eta\,\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}. -\end{equation} +\textbf{Tangent space --- linear approximation at \(p\):} -This is \emph{invariant to reparametrisation} and achieves quadratic convergence -on convex objectives — equivalent to Fisher scoring. +\begin{verbatim} + M (curved 2D surface): TpM (flat tangent plane at p): -\textbf{KL geometry:} -\begin{equation} - \KL{p_\theta}{p_{\theta+d\theta}} - = \tfrac12\,d\theta^\top\mathcal{I}(\theta)\,d\theta + O(\|d\theta\|^3), -\end{equation} -confirming Fisher--Rao as the intrinsic KL metric. + .~~~~. ___________ + / \ --> | TpM | + | p * | | * p | + | | |___________| + \ / + .~~~~. + (not flat globally, but TpM is flat locally — used for calculus on M) +\end{verbatim} -\textbf{Dually flat structure:} Exponential families -$p(x;\theta)=h(x)\exp(\theta^\top T(x)-A(\theta))$ -are $e$-flat in natural parameters and $m$-flat in mean parameters -$\eta=\nabla A(\theta)$, with vanishing sectional curvature $K=0$ — -explaining exact Newton/natural-gradient convergence on these models. +\textbf{Geodesics} satisfy: -\subsection{Lie Groups and Geometric Control} +\[\ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0,\] -\begin{defbox}{Lie Group} -A \emph{Lie group} $G$ is a smooth manifold with a group structure where -multiplication and inversion are smooth. The \emph{Lie algebra} $\mathfrak{g} = T_e G$ +where \(\Gamma^k_{ij} = \tfrac12 g^{kl}(\partial_i g_{jl}+\partial_j g_{il}-\partial_l g_{ij})\) are the \emph{Christoffel symbols} encoding intrinsic curvature. + +\subsubsection{10.2 Information Geometry and Fisher-Rao Metric}\label{information-geometry-and-fisher-rao-metric} + +The statistical manifold \(\mathcal{M} = \{p(\cdot;\theta)\}\) carries the \textbf{Fisher-Rao metric} \(g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}\). + +\textbf{Standard vs natural gradient:} + +\begin{verbatim} +Standard gradient descent: Natural gradient descent: + theta_{k+1} = theta_k - eta * grad L theta_{k+1} = theta_k - eta * I^{-1} grad L + + Parameter space = flat R^d. Parameter space = Riemannian (metric I(theta)). + Ignores curvature. Adapts step to local geometry. + Slow on ill-conditioned I. Invariant to reparametrisation. + O(kappa(I)) iterations. O(1) iterations on exponential families. +\end{verbatim} + +\textbf{Natural gradient (Amari 1998):} + +\[\theta \leftarrow \theta - \eta\,\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}.\] + +\textbf{KL geometry:} \(D_{\mathrm{KL}}(p_\theta\,\|\,p_{\theta+d\theta}) = \tfrac12\,d\theta^\top\mathcal{I}(\theta)\,d\theta + O(\|d\theta\|^3)\), confirming Fisher-Rao as the intrinsic KL metric. + +\textbf{Dually flat structure:} Exponential families \(p(x;\theta)=h(x)\exp(\theta^\top T(x)-A(\theta))\) have \(K=0\) --- explaining exact Newton/natural-gradient convergence. + +\begin{tcolorbox}[colback=notebg!60,colframe=optgreen,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Example}} — Example — Natural Gradient on a Gaussian Model},breakable] +For $p(x;\theta) = \mathcal{N}(\mu, \sigma^2)$, $\theta=(\mu,\sigma^2)$: + +$$\mathcal{I}(\theta) = \begin{pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4) \end{pmatrix}.$$ + +**Natural gradient** of $\mathcal{L} = -\log p(x_{\rm obs};\theta)$: + +$$\tilde\nabla_\theta\mathcal{L} = \mathcal{I}^{-1}\nabla\mathcal{L} = \begin{pmatrix}\mu-x \\ \sigma^2 - (x-\mu)^2/2\end{pmatrix}.$$ + +One Newton step on this exponential family finds the MLE exactly because the +Hessian equals $\mathcal{I}$ (dually flat, $K=0$). +\end{tcolorbox} + +\subsubsection{10.3 Lie Groups and Geometric Control}\label{lie-groups-and-geometric-control} + +\begin{tcolorbox}[colback=defbg!60,colframe=optblue,fonttitle=\bfseries\sffamily,title={\textsf{\textbf{Definition}} — Definition — Lie Group},breakable] +A *Lie group* $G$ is a smooth manifold with a group structure where +multiplication and inversion are smooth. The *Lie algebra* $\mathfrak{g} = T_e G$ linearises the group at the identity. -\end{defbox} +\end{tcolorbox} -\textbf{Examples:} $SO(d)$ (rotation group), $GL(d)$ (invertible matrices), -Heisenberg group (path-signature features). +\textbf{Matrix Lie group hierarchy:} -\textbf{Left-invariant control system on $G$:} -\begin{equation} - \dot{g}(t) = g(t)\,\xi(t), \quad g\in G,\; \xi(t)\in\mathfrak{g}. -\end{equation} +\begin{verbatim} +GL(n,R) general linear group (all invertible n x n real matrices) + | + |--- SL(n,R) special linear (det = 1) + | + |--- O(n) orthogonal (R'R = I) + | | + | +---- SO(n) special orthogonal (det = +1, pure rotations) + | Used in: portfolio factor rotation, PCA constraints + | + +--- Sp(2n,R) symplectic (preserves symplectic form omega) + Used in: Hamiltonian mechanics, PMP sections 4.2 and 10.4 -PMP on Lie groups yields the \emph{Lie--Poisson (Euler--Poincaré) equations} -(Holm--Marsden--Ratiu), providing structure-preserving optimal trajectories. +Heisenberg group H(n): upper triangular with 1s on diagonal. + Used in: path-signature feature maps (lab_signature_methods) +\end{verbatim} -\subsection{Symplectic Geometry and Hamiltonian Structure} -\label{sec:symplectic} +\textbf{Left-invariant control system on \(G\):} -The phase space $(T^\star M, \omega)$ carries the symplectic 2-form -$\omega = \sum_i dp_i \wedge dq_i$. Hamilton's equations preserve $\omega$ -(\emph{Liouville's theorem} — phase-space volume conserved). +\[\dot g(t) = g(t)\,\xi(t), \quad g\in G,\; \xi(t)\in\mathfrak{g}.\] -\textbf{Connection to PMP:} The costate pair $(X_t^\star, p_t)$ solves -Hamilton's equations, i.e., the PMP is a symplectic flow on $T^\star\R^d$. +PMP on Lie groups yields the \emph{Lie-Poisson (Euler-Poincare) equations} (Holm-Marsden-Ratiu), providing structure-preserving optimal trajectories. -\textbf{Symplectic integrators} (Störmer--Verlet, Ruth--Forest) preserve $\omega$ -discretely, keeping the Hamiltonian nearly constant over long horizons. +\subsubsection{10.4 Symplectic Geometry and Hamiltonian Structure}\label{symplectic-geometry-and-hamiltonian-structure} -\subsection{Sectional Curvature and Landscape Geometry} +The phase space \((T^\star M, \omega)\) carries the symplectic 2-form \(\omega = \sum_i dp_i \wedge dq_i\). Hamilton's equations preserve \(\omega\) (\emph{Liouville's theorem} --- phase-space volume conserved). -The sectional curvature $K(\sigma)$ governs how quickly nearby geodesics diverge: +\textbf{Connection to PMP:} The costate pair \((X_t^\star, p_t)\) solves Hamilton's equations, i.e., the PMP is a symplectic flow on \(T^\star\mathbb{R}^d\). -\begin{center} -\begin{tabular}{L{3cm} L{5cm} L{5cm}} -\toprule -$K>0$ (sphere) & Geodesics converge & Compact optimiser trajectories \\ -$K=0$ (flat) & Euclidean behaviour & Newton / natural gradient exact \\ -$K<0$ (hyperbolic) & Exponential spread & Efficient landscape exploration \\ -\bottomrule -\end{tabular} -\end{center} +\textbf{Symplectic integrators} (Stormer-Verlet, Ruth-Forest) preserve \(\omega\) discretely, keeping the Hamiltonian nearly constant over long horizons --- critical for multi-year allocation back-tests in Optimiz-rs. -For exponential families in natural/mean parameters, $K=0$ — explaining exact -Newton convergence without curvature correction. +\subsubsection{10.5 Sectional Curvature and Landscape Geometry}\label{sectional-curvature-and-landscape-geometry} -%% ═══════════════════════════════════════════════════════════════════════════ -\section*{Quick Reference} -%% ═══════════════════════════════════════════════════════════════════════════ -\addcontentsline{toc}{section}{Quick Reference} +The sectional curvature \(K(\sigma)\) governs how quickly nearby geodesics diverge: -\begin{center} -\small -\begin{tabularx}{\linewidth}{L{3.8cm} X L{3.2cm}} -\toprule -\textbf{Concept} & \textbf{Key equation / object} & \textbf{Module} \\ -\midrule -Brownian motion & $W_t - W_s \sim \mathcal{N}(0,t-s)$ & \texttt{point\_processes} \\ -Itô SDE & $dX=b\,dt+\sigma\,dW$ & \texttt{ou\_estimator} \\ -Poisson / Compound Poisson& $N_t\sim\text{Poisson}(\lambda t)$ & \texttt{point\_processes} \\ -Lévy process & triplet $(b,\sigma^2,\nu)$ & \texttt{point\_processes} \\ -HJB PDE & $-\partial_t V = \inf_u[\ell + \nabla V^\top b + \tfrac12\tr\sigma\sigma^\top\nabla^2 V]$ & \texttt{optimal\_control} \\ -HJBI (jumps) & $+\int[V(\cdot+c)-V-\nabla V^\top c]\nu\,dz$ & \texttt{optimal\_control} \\ -PMP costate & $\dot{p} = -\nabla_x\mathcal{H}$,\; $u^\star=\arg\min_u\mathcal{H}$ & \texttt{optimal\_control} \\ -MFG (HJB + KFP) & fixed-point $u,m$ & \texttt{mean\_field\_games} \\ -Kalman filter & $K_t = P^-H^\top(HP^-H^\top+R)^{-1}$ & \texttt{optimal\_control} \\ -MALA & $x'=x-\tfrac{h^2}{2}\nabla U+h\xi$ & \texttt{mcmc} \\ -HMM & Baum--Welch EM + Viterbi & \texttt{hmm} \\ -Fisher information & $\mathcal{I}_{ij}=\E[\partial_i\ell\,\partial_j\ell]$ & \texttt{hmm}, \texttt{sparse} \\ -Natural gradient & $\mathcal{I}^{-1}\nabla_\theta\mathcal{L}$ & \texttt{differential\_evolution} \\ -Riemannian / Lie geometry & Christoffel symbols, Lie--Poisson eqs & experimental \\ -DE (jDE) & mutation + crossover + selection & \texttt{differential\_evolution} \\ -\bottomrule -\end{tabularx} -\end{center} +\begin{verbatim} +K > 0 (sphere): geodesics converge -> compact optimiser trajectories +K = 0 (flat ): Euclidean behaviour -> Newton / natural gradient exact +K < 0 (hyper.): exponential spread -> efficient landscape exploration +\end{verbatim} -%% ═══════════════════════════════════════════════════════════════════════════ -\section*{References} -%% ═══════════════════════════════════════════════════════════════════════════ -\addcontentsline{toc}{section}{References} +For exponential families in natural/mean parameters \(K=0\) --- explaining exact Newton convergence without curvature correction. -\begin{enumerate}[label={[\arabic*]}, leftmargin=2em] - \item Øksendal, B. \textit{Stochastic Differential Equations}, 6th ed. Springer, 2003. - \item Cont, R. \& Tankov, P. \textit{Financial Modelling with Jump Processes}. CRC Press, 2004. - \item Fleming, W.H. \& Soner, H.M. \textit{Controlled Markov Processes and Viscosity Solutions}. Springer, 2006. - \item Lasry, J.-M. \& Lions, P.-L. ``Mean field games.'' \textit{Jpn.\ J.\ Math.}\ \textbf{2} (2007) 229--260. - \item Amari, S. \textit{Information Geometry and Its Applications}. Springer, 2016. - \item do Carmo, M.P. \textit{Riemannian Geometry}. Birkhäuser, 1992. - \item Holm, D.D., Marsden, J.E. \& Ratiu, T.S. ``The Euler--Poincaré equations.'' \textit{Adv.\ Math.}\ \textbf{137} (1998). - \item Price, K.V., Storn, R.M. \& Lampinen, J.A. \textit{Differential Evolution}. Springer, 2005. - \item Roberts, G.O., Gelman, A. \& Gilks, W.R. ``Weak convergence of Metropolis algorithms.'' (1997). - \item Merton, R.C. ``Option pricing when underlying stock returns are discontinuous.'' \textit{JFE}\ \textbf{3} (1976). - \item Crandall, M.G. \& Lions, P.-L. ``Viscosity solutions of Hamilton--Jacobi equations.'' \textit{Trans.\ AMS} (1983). +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{Quick Reference}\label{quick-reference} + +{\def\LTcaptype{none} % do not increment counter +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.1800}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.4400}} + >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3800}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Concept +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Key equation / object +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Optimiz-rs module +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Brownian motion & \(W_t - W_s \sim \mathcal{N}(0,t-s)\) & \texttt{point\_processes} \\ +Ito SDE & \(dX=b\,dt+\sigma\,dW\) & \texttt{ou\_estimator} \\ +Poisson / Compound Poisson & \(N_t\sim\text{Poisson}(\lambda t)\) & \texttt{point\_processes} \\ +Levy process & triplet \((b,\sigma^2,\nu)\) & \texttt{point\_processes} \\ +HJB PDE & \(-\partial_t V = \inf_u[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}\sigma\sigma^\top\nabla^2 V]\) & \texttt{optimal\_control} \\ +HJBI (jumps) & \(+\int[V(\cdot+c)-V-\nabla V^\top c]\nu\,dz\) & \texttt{optimal\_control} \\ +PMP costate & \(\dot p = -\nabla_x\mathcal{H}\), \(u^\star=\arg\min_u\mathcal{H}\) & \texttt{optimal\_control} \\ +MFG (HJB + KFP) & fixed-point \(u,m\) & \texttt{mean\_field\_games} \\ +Kalman filter & \(K_t = P^-H^\top(HP^-H^\top+R)^{-1}\) & \texttt{optimal\_control} \\ +MALA & \(x'=x-\tfrac{h^2}{2}\nabla U+h\xi\) & \texttt{mcmc} \\ +HMM & Baum-Welch EM + Viterbi & \texttt{hmm} \\ +Fisher information & \(\mathcal{I}_{ij}=\mathbb{E}[\partial_i\ell\,\partial_j\ell]\) & \texttt{hmm}, \texttt{sparse} \\ +Natural gradient & \(\mathcal{I}^{-1}\nabla_\theta\mathcal{L}\) & \texttt{differential\_evolution} \\ +Riemannian / Lie geometry & Christoffel symbols, Lie-Poisson equations & experimental \\ +DE (jDE) & mutation + crossover + selection & \texttt{differential\_evolution} \\ +\end{longtable} +} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\subsection{References}\label{references} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} + +\item + Oksendal, B. \emph{Stochastic Differential Equations}, 6th ed.~Springer, 2003. +\item + Cont, R. \& Tankov, P. \emph{Financial Modelling with Jump Processes}. CRC Press, 2004. +\item + Fleming, W.H. \& Soner, H.M. \emph{Controlled Markov Processes and Viscosity Solutions}. Springer, 2006. +\item + Lasry, J.-M. \& Lions, P.-L. ``Mean field games.'' \emph{Jpn. J. Math.} \textbf{2} (2007) 229-260. +\item + Amari, S. \emph{Information Geometry and Its Applications}. Springer, 2016. +\item + do Carmo, M.P. \emph{Riemannian Geometry}. Birkhauser, 1992. +\item + Holm, D.D., Marsden, J.E. \& Ratiu, T.S. ``The Euler-Poincare equations.'' \emph{Adv. Math.} \textbf{137} (1998). +\item + Price, K.V., Storn, R.M. \& Lampinen, J.A. \emph{Differential Evolution}. Springer, 2005. +\item + Roberts, G.O., Gelman, A. \& Gilks, W.R. ``Weak convergence of Metropolis algorithms.'' (1997). +\item + Merton, R.C. ``Option pricing when underlying stock returns are discontinuous.'' \emph{JFE} \textbf{3} (1976). +\item + Crandall, M.G. \& Lions, P.-L. ``Viscosity solutions of Hamilton-Jacobi equations.'' \emph{Trans. AMS} (1983). +\item + Almgren, R. \& Chriss, N. ``Optimal execution of portfolio transactions.'' \emph{J. Risk} \textbf{3} (2001). +\item + Carmona, R. \& Delarue, F. \emph{Probabilistic Theory of Mean Field Games}. Springer, 2018. \end{enumerate} \end{document} diff --git a/docs/source/conf.py b/docs/source/conf.py index 7f98ec4..6158b36 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -35,8 +35,8 @@ html_theme = 'furo' # Modern, clean theme html_static_path = ['_static'] html_title = 'Optimiz-rs Documentation' html_short_title = 'Optimiz-rs' -html_logo = 'logo_optimizrs.png' -html_favicon = 'logo_optimizrs.png' +html_logo = 'logo_optimizrs_transparent.png' +html_favicon = 'logo_optimizrs_transparent.png' html_theme_options = { "light_css_variables": { diff --git a/docs/source/logo_optimizrs_transparent.png b/docs/source/logo_optimizrs_transparent.png new file mode 100644 index 0000000000000000000000000000000000000000..e6fc400f7be93ae921d4f21746ece339372090df GIT binary patch literal 1365855 zcmV(nK=QwdP)$VQfJO3GuA*!Z@Z?edZ3{ zEkxJzKEK7Oq;UpsC3<<^g@}2Slc|&EZ2K@O3DMnaUj|j)pH4jAG};W&xZ7VgZJdqL zpWE@H1O1FZ&NNdcwAPg$roB7B4-wbZv1C3UgZE-(1jB)SNb^lZR}|Z4^u3XGq9s(z 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+2,8 @@ This page develops the core mathematics underlying Optimiz-rs's Rust kernels — from first principles through advanced theory. Each section opens with a **definition block**, builds -intuition through **examples**, and closes with a **notebook micro-check**. For complete -walkthroughs see `examples/notebooks/`. +intuition through **examples and diagrams**, and closes with a **notebook micro-check**. +For complete walkthroughs see `examples/notebooks/`. --- @@ -19,7 +19,29 @@ solutions $\{\mathbf{x}_{i,g}\} \subset \mathbb{R}^d$. unbiased directional finite-difference of $f$, so DE implicitly estimates curvature without Jacobians. -### Operators +### 1.1 Geometric Intuition — Mutation in $\mathbb{R}^2$ + +``` + x_r3 * + | + | F*(x_r2 - x_r3) F = scale factor in [0,2] + | ------------------> + x_r2 * * v_i (mutant) + \\ / + \\________________/ + (difference vec) + + x_r1 *------------------------------------------------> v_i + base vector mutation vector added +``` + +- $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3$ are three **distinct** randomly selected parents. +- The mutant $\mathbf{v}_i$ lands on the other side relative to $\mathbf{x}_{r_1}$. +- **Crossover** then mixes $\mathbf{v}_i$ and $\mathbf{x}_i$ dimension-by-dimension with + probability $CR$, producing trial vector $\mathbf{u}_i$. +- **Selection** keeps $\mathbf{u}_i$ only if it improves over $\mathbf{x}_i$ — pure greedy. + +### 1.2 Operators | Step | Formula | Role | |------|---------|------| @@ -30,7 +52,7 @@ without Jacobians. **Convergence (informal):** Under bounded population diversity and Lipschitz $f$, the best-so-far value converges a.s. to a stationary point as $N,g\to\infty$ (Price et al. 2005). -### Self-Adaptive jDE (Optimiz-rs default) +### 1.3 Self-Adaptive jDE (Optimiz-rs default) Parameters $F,CR$ are per-individual and reset stochastically each generation: @@ -43,6 +65,54 @@ $$ $\tau_1=\tau_2=0.1$ by default. On rugged landscapes this produces bimodal $F$ histograms concentrated near 0.8 — a sign the landscape is highly multimodal. +### 1.4 Example — Minimising the Rastrigin Function + +The Rastrigin function $f(\mathbf{x}) = 10d + \sum_i[x_i^2 - 10\cos(2\pi x_i)]$ +has $\approx 10^d$ local minima (global minimum $f^*=0$ at $\mathbf{x}^*=\mathbf{0}$). + +**Why gradient methods fail:** The gradient $\partial_{x_i}f = 2x_i + 20\pi\sin(2\pi x_i)$ +oscillates rapidly — any gradient step hops between basins. + +**Why DE succeeds:** The difference vector $F(\mathbf{x}_{r_2}-\mathbf{x}_{r_3})$ +spans the characteristic basin width (~1.0), enabling inter-basin jumps. + +``` +Rastrigin 1D sketch (d=1): + + f(x) + | * * * <- local minima (many) + | * * * * * * + |* * + | * * + +------|------+------|-- x + -1 0 1 + ^ + global min f=0 +``` + +**Typical jDE convergence** ($d=10$, $N=100$, $\tau_1=\tau_2=0.1$): + +``` +Gen Best f Mean F Mean CR +---- ------- ------- ------- + 1 48.3 0.50 0.50 + 50 12.1 0.78 0.31 +200 3.4 0.82 0.24 <- F clusters near 0.8 (bimodal) +500 0.0 0.83 0.22 <- converged +``` + +::::{admonition} Tip — Diagnosing Stagnation +:class: tip + +If best-$f$ does not decrease for 100+ generations: + +1. **Check $F$ histogram.** Bimodal near 0.8 -> landscape is multimodal (increase $N$). + Collapsed near 0 -> diversity loss; restart with random perturbation. +2. **Check $CR$ distribution.** Uniform -> dimensions not interacting. + Collapsed near 0 -> DE treating dimensions independently (separable function). +3. **Increase $N$** to $\approx 10d$ for $d > 20$. +:::: + **Notebook check** (`05_performance_benchmarks.ipynb`): Plot $F_i, CR_i$ histograms every 50 generations; expect values clustering in $[0.5,0.9]$ on hard problems. @@ -51,9 +121,42 @@ every 50 generations; expect values clustering in $[0.5,0.9]$ on hard problems. ## 2 · Stochastic Processes These form the probabilistic backbone of all continuous-time models in Optimiz-rs. +We build the theory from scratch: random walk → Brownian motion → Itō calculus → SDEs → jump-diffusions. + +--- ### 2.1 Brownian Motion +#### 2.1.0 Intuitive Construction — From Random Walk to BM + +**Step 1 — Discrete random walk.** Flip a fair coin $n$ times per unit time. +Define $\xi_k = +1$ (heads) or $-1$ (tails) i.i.d. After $n$ steps of size $1/\sqrt{n}$: + +$$S^{(n)}_t = \frac{1}{\sqrt{n}}\sum_{k=1}^{\lfloor nt \rfloor} \xi_k.$$ + +By the **Central Limit Theorem**, as $n\to\infty$: $S^{(n)}_t \xrightarrow{d} W_t \sim \mathcal{N}(0,t)$. + +``` +Coin-flip random walk (n=20 steps per unit time): + + W_t ++2 | * * + | * * * + 0 | * * * * * + |* * * +-2 | * + +----------------------------> t + 0 0.5 1.0 + + n → ∞ ("zoom out"): jagged → smooth BM fan +``` + +**Step 2 — Scaling limit.** The normalization $1/\sqrt{n}$ is crucial: +- Without it, variance grows as $n$ (diverges). +- With $n^{-1/2}$: variance = $n \cdot (1/\sqrt{n})^2 \cdot t = t$ — exactly right. + +This is why $W_t \sim \mathcal{N}(0,t)$: **variance accumulates linearly in time**. + ::::{admonition} Definition — Wiener Process :class: definition @@ -66,15 +169,67 @@ is a *standard Brownian motion* if: 4. Paths $t\mapsto W_t(\omega)$ are **continuous** a.s. :::: -**Key properties:** +#### 2.1.1 Key Analytical Properties -- $\mathbb{E}[W_t] = 0$, $\operatorname{Var}(W_t) = t$, $\operatorname{Cov}(W_s,W_t) = \min(s,t)$. -- **Quadratic variation:** $[W]_T = T$ (paths are non-differentiable but have finite $p$-variation for $p>2$). -- **Self-similarity:** $c^{-1/2}W_{ct} \overset{d}{=} W_t$ (Hurst exponent $H=\tfrac12$). +| Property | Formula | Intuition | +|----------|---------|-----------| +| Mean | $\mathbb{E}[W_t] = 0$ | No drift — symmetric random walk | +| Variance | $\operatorname{Var}(W_t) = t$ | Uncertainty grows with time | +| Covariance | $\operatorname{Cov}(W_s,W_t) = \min(s,t)$ | Shared history up to first time | +| Non-differentiability | $\lim_{h\to 0}(W_{t+h}-W_t)/h$ diverges a.s. | Too "rough" for ordinary calculus | +| Quadratic variation | $[W]_T = T$ | Core source of Itō correction term | +| Self-similarity | $c^{-1/2}W_{ct} \overset{d}{=} W_t$ | Fractal structure, Hurst $H=\tfrac12$ | + +**Quadratic variation derivation (step by step):** + +Partition $[0,T]$ into $n$ pieces of width $\Delta = T/n$. Sum of squared increments: + +$$\sum_{k=0}^{n-1}(W_{t_{k+1}}-W_{t_k})^2 \overset{?}{=} T \quad \text{as } n\to\infty.$$ + +**Step 1** — Each increment: $(W_{t_{k+1}}-W_{t_k})^2 \sim \Delta \cdot \chi_1^2$, so +$\mathbb{E}[(W_{t_{k+1}}-W_{t_k})^2] = \Delta$. + +**Step 2** — Sum of means: $\sum_{k=0}^{n-1} \Delta = n\Delta = T$. + +**Step 3** — Variance of the sum: +$\operatorname{Var}\!\left(\sum (W_{t_{k+1}}-W_{t_k})^2\right) = n \cdot 2\Delta^2 = 2T^2/n \xrightarrow{n\to\infty} 0$. + +**Conclusion:** $\sum (W_{t_{k+1}}-W_{t_k})^2 \xrightarrow{L^2} T$. We write $dW_t^2 = dt$. +This **single identity** is the engine of all Itō calculus. + +::::{admonition} Why dW² = dt is remarkable +:class: tip + +In ordinary calculus, $dx^2 \approx dx \cdot dx \to 0$ (second-order infinitesimal). +For Brownian motion, $(dW)^2 = dt$ is **first-order** — it does **not** vanish! + +Physically: BM paths oscillate so rapidly ($\sim t^{0.5}$ scale) that their squared +increments accumulate at rate $1$ — comparable to the drift $dt$. + +This is the **only** reason Itō's lemma has an extra term. +:::: + +**Multiple sample paths** — the fan widens as $\propto\sqrt{t}$: + +``` +W_t + +2 | ........... + | .... ..... + 0 |.. ..... <- E[W_t] = 0 (all paths centered) + | ....--...... + -2 | ........ + +--------------------------> t + 0 T + + Width ~ 2*sqrt(t) (95% of paths stay within +/- 2*sqrt(t)) + + Each dot is a BM path — different because different coin flips. + As T grows, the "trumpet" opens wider. +``` **Example — Geometric BM:** $S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)$ -is the Black–Scholes price model. Sample path sketch: +is the Black-Scholes price model. Log-normal marginals; continuous, nowhere-differentiable paths: ``` S_t @@ -84,52 +239,331 @@ S_t |/ +-------------------------------> t 0 T -(log-normal marginals; continuous, nowhere-differentiable paths) ``` -### 2.2 Itô Calculus +**Multiple sample paths** — the fan widens as $\propto\sqrt{t}$: -::::{admonition} Definition — Itô Integral +``` +W_t + +2 | ........... + | .... ..... + 0 |.. ..... <- E[W_t] = 0 (all paths centered) + | ....--...... + -2 | ........ + +--------------------------> t + 0 T + + Width grows as sqrt(t) (68% of paths stay within +/- sqrt(t)) +``` + +**Example — Geometric BM:** +$S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)$ +is the Black-Scholes price model. Log-normal marginals; continuous, nowhere-differentiable paths: + +``` +S_t +| .---. +| .--./ \\----. +| / \\---------. +|/ ++-------------------------------> t + 0 T +``` + +### 2.2 Itō Calculus + +#### 2.2.0 Why You Cannot Use Ordinary Integration + +Attempt to define $\int_0^T W_t\,dW_t$ using a Riemann sum: pick $W_{t_k}$ at the +**left endpoint** → get one answer; pick $(W_{t_k}+W_{t_{k+1}})/2$ (midpoint) → get a *different* answer. + +This ambiguity occurs because $W$ is not of bounded variation. **Itō's convention** +(left endpoint) is the only one that produces a **martingale** — ensuring no look-ahead. + +::::{admonition} Definition — Itō Integral :class: definition For adapted $f \in \mathcal{L}^2$ (i.e. $\mathbb{E}\!\int_0^T f_t^2\,dt < \infty$): -$$\int_0^T f_t\,dW_t \;=\; L^2\text{-}\lim_{|\pi|\to 0} \sum_{k} f_{t_k}(W_{t_{k+1}}-W_{t_k}).$$ +$$\int_0^T f_t\,dW_t \;:=\; L^2\text{-}\lim_{|\pi|\to 0} \sum_{k} f_{t_k}(W_{t_{k+1}}-W_{t_k}).$$ -The Itô integral is a **martingale** with zero mean and **Itô isometry** -$\mathbb{E}\bigl[(\int_0^T f_t\,dW_t)^2\bigr] = \mathbb{E}\int_0^T f_t^2\,dt$. +Key guarantees: +- **Zero mean:** $\mathbb{E}\!\left[\int_0^T f_t\,dW_t\right] = 0$. +- **Itō isometry:** $\mathbb{E}\!\left[\left(\int_0^T f_t\,dW_t\right)^2\right] = \mathbb{E}\!\int_0^T f_t^2\,dt$. +- **Martingale:** $M_t = \int_0^t f_s\,dW_s$ satisfies $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$. :::: -::::{admonition} Theorem — Itô's Lemma +**Itō isometry — proof sketch:** +Let $I_T = \sum_k f_{t_k}\Delta W_k$ (simple process). Then: + +$$\mathbb{E}[I_T^2] = \sum_{j,k}\underbrace{\mathbb{E}[f_{t_j}\Delta W_j \cdot f_{t_k}\Delta W_k]}_{\text{cross terms}}$$ + +For $j \neq k$ (say $j < k$): $f_{t_j}\Delta W_j$ and $f_{t_k}$ are both $\mathcal{F}_{t_k}$-measurable, +while $\Delta W_k$ is **independent** of $\mathcal{F}_{t_k}$ with mean 0 → cross term $= 0$. + +For $j = k$: $\mathbb{E}[f_{t_j}^2 (\Delta W_j)^2] = \mathbb{E}[f_{t_j}^2]\Delta t_j$ (independence of $f_{t_j}$ and $\Delta W_j$). + +$$\Rightarrow \mathbb{E}[I_T^2] = \sum_k \mathbb{E}[f_{t_k}^2]\Delta t_k \xrightarrow{|\pi|\to 0} \mathbb{E}\int_0^T f_t^2\,dt. \quad \checkmark$$ + +#### 2.2.1 Itō's Lemma — Full Derivation + +::::{admonition} Theorem — Itō's Lemma :class: tip For $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and $F \in C^{1,2}([0,T]\times\mathbb{R})$: -$$dF(t,X_t) = \partial_t F\,dt + \partial_x F\,dX_t + \tfrac{1}{2}\partial_{xx}F\,\sigma_t^2\,dt.$$ +$$\boxed{dF(t,X_t) = \partial_t F\,dt + \partial_x F\,dX_t + \tfrac{1}{2}\sigma_t^2\,\partial_{xx}F\,dt}$$ -The correction term $\tfrac12\sigma^2\partial_{xx}F$ (absent in ordinary calculus) arises -from the non-zero quadratic variation $d[W]_t = dt$. +Expanded: + +$$dF = \underbrace{\left(\partial_t F + \mu_t\,\partial_x F + \tfrac12\sigma_t^2\,\partial_{xx}F\right)}_{\text{drift}}\,dt ++ \underbrace{\sigma_t\,\partial_x F}_{\text{diffusion}}\,dW_t.$$ :::: -**Example:** Let $X_t = \log S_t$ with $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$. -Itô's Lemma gives $dX_t = (\mu - \tfrac12\sigma^2)\,dt + \sigma\,dW_t$. ✔ +**Derivation — Taylor expand $F(t+dt, X_{t+dt})$:** -### 2.3 General Itô SDEs +$$dF = \partial_t F\,dt + \partial_x F\,dX + \tfrac12\partial_{xx}F\,(dX)^2 + \underbrace{\partial_{tx}F\,dt\,dX + \ldots}_{\to 0} $$ + +Compute $(dX)^2$ using the **Itō multiplication table**: + +| × | $dt$ | $dW_t$ | +|---|------|--------| +| $dt$ | $0$ | $0$ | +| $dW_t$ | $0$ | $dt$ | + +$$\begin{aligned} +(dX_t)^2 &= (\mu_t\,dt + \sigma_t\,dW_t)^2 \\ + &= \mu_t^2\underbrace{(dt)^2}_{0} + 2\mu_t\sigma_t\underbrace{dt\cdot dW_t}_{0} + \sigma_t^2\underbrace{(dW_t)^2}_{dt}\\ + &= \sigma_t^2\,dt. +\end{aligned}$$ + +Substituting: + +$$dF = \partial_t F\,dt + \partial_x F(\mu_t\,dt + \sigma_t\,dW_t) + \tfrac12\partial_{xx}F\cdot\sigma_t^2\,dt$$ + +$$= \left(\partial_t F + \mu_t\partial_x F + \tfrac12\sigma_t^2\partial_{xx}F\right)dt + \sigma_t\partial_x F\,dW_t. \quad \checkmark$$ + +**The extra term $\tfrac12\sigma^2\partial_{xx}F\,dt$ is the "Itō correction".** +In ordinary calculus $(dW)^2=0$, so it vanishes. In stochastic calculus, BM oscillates +so rapidly that $(dW)^2 = dt$ — a first-order effect. + +**Multidimensional version** (for vector $\mathbf{X}\in\mathbb{R}^n$, matrix noise): + +$$dF = \partial_t F\,dt + \sum_i \partial_{x_i}F\,dX_i + \tfrac12\sum_{i,j}\partial_{x_ix_j}F\,d[X_i,X_j]_t$$ + +where $d[X_i, X_j]_t = d\langle X_i, X_j\rangle_t$ is the quadratic co-variation. + +#### 2.2.2 Worked Examples of Itō's Lemma + +**Example 1 — GBM, derive explicit solution:** + +SDE: $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$. + +**Goal:** Find $S_t$ in closed form. + +**Step 1** — Guess $F(t,x) = \log x$. Compute partials: +$\partial_t F = 0$, $\partial_x F = 1/x$, $\partial_{xx}F = -1/x^2$. + +**Step 2** — Apply Itō's lemma: + +$$d(\log S_t) = 0 + \frac{1}{S_t}\,dS_t + \tfrac12\cdot(-\tfrac{1}{S_t^2})\cdot\sigma^2 S_t^2\,dt$$ + +$$= \frac{\mu S_t\,dt + \sigma S_t\,dW_t}{S_t} - \tfrac12\sigma^2\,dt$$ + +$$= \left(\mu - \tfrac12\sigma^2\right)\,dt + \sigma\,dW_t.$$ + +**Step 3** — Integrate (deterministic integral + Itō integral): + +$$\log S_T - \log S_0 = \left(\mu-\tfrac12\sigma^2\right)T + \sigma W_T.$$ + +**Step 4** — Exponentiate: + +$$\boxed{S_T = S_0\exp\!\left[\left(\mu - \tfrac12\sigma^2\right)T + \sigma W_T\right].}$$ + +The **Itō correction** $-\tfrac12\sigma^2 T$ lowers the expected log-return: +$\mathbb{E}[\log S_T] = \log S_0 + (\mu-\tfrac12\sigma^2)T$, +but $\mathbb{E}[S_T] = S_0 e^{\mu T}$ (Jensen's inequality explains the gap: +$e^{\mathbb{E}[X]} < \mathbb{E}[e^X]$ for non-degenerate $X$). + +``` +log S_t +| Ito correction: slope = mu - sigma^2/2 (lower than mu!) +| _.-' +| / <- Without Ito correction, slope would be mu (wrong!) +|/ ++----------------------------> t + +E[log S_t] = log S_0 + (mu - sigma^2/2)*t + (time-averaged growth, always less than mu for sigma > 0) +``` + +**Example 2 — Itō product rule ($d(X_t Y_t)$):** + +By Itō's lemma applied to $F(x,y) = xy$: + +$$d(X_t Y_t) = Y_t\,dX_t + X_t\,dY_t + d[X,Y]_t$$ + +where $d[X,Y]_t = \sigma_X\sigma_Y\,dt$. Compare to ordinary calculus: $d(xy) = y\,dx + x\,dy$ (no cross term because $(dx)^2=0$). + +**Example 3 — Integration by parts for stochastic integrals:** + +$$\int_0^T W_t\,dW_t = \tfrac12 W_T^2 - \tfrac12 T.$$ + +Ordinary calculus would give $\int_0^T W_t\,dW_t = \tfrac12 W_T^2$. +The $-\tfrac12 T$ correction comes from the quadratic variation. + +**Verification via Itō's lemma:** Set $F(t,x) = x^2/2$: +$dF = x\,dW + \tfrac12\cdot 1 \cdot dt = W_t\,dW_t + \tfrac12\,dt$. +Integrate: $\tfrac12 W_T^2 - 0 = \int_0^T W_t\,dW_t + \tfrac12 T$ → result follows. ✓ + +#### 2.2.3 Itō vs Stratonovich + +| Property | Itō integral | Stratonovich integral ($\circ$) | +|----------|-------------|----------------------| +| Chain rule | Modified ($+\tfrac12\sigma^2\partial_{xx}F$ term) | Standard calculus chain rule | +| Martingale | Yes (if $f$ adapted) | No in general | +| Use in finance | Natural (no look-ahead) | Physics, geometry | +| Conversion | $\int f\circ dW = \int f\,dW + \tfrac12\int \partial_x f\,\sigma\,dt$ | (same identity) | +| SDE solutions | Different numerics needed | Standard ODE methods work | + +**Conversion formula** — Itō $\to$ Stratonovich: + +$$\int_0^T f(X_t)\circ dW_t = \int_0^T f(X_t)\,dW_t + \tfrac{1}{2}\int_0^T f'(X_t)\sigma_t\,dt.$$ + +**Rule of thumb:** Use Itō in finance (causality, no-arbitrage); use Stratonovich in +physics/differential geometry (coordinate-invariant chain rule). + +### 2.3 General Itō SDEs $$dX_t = b(t, X_t)\,dt + \boldsymbol{\sigma}(t, X_t)\,dW_t,\quad X_0 = x_0.$$ -**Existence & uniqueness (Picard–Lindelöf for SDEs):** If $b, \boldsymbol{\sigma}$ are -globally Lipschitz with linear growth, there exists a unique strong solution with -$\mathbb{E}[\sup_{t\le T}\|X_t\|^2]<\infty$. +#### 2.3.0 Existence, Uniqueness and Picard Iteration -**Common SDE Models** +::::{admonition} Theorem — Strong Solution Existence (Picard–Lindelöf for SDEs) +:class: tip -| Process | SDE | Stationary distribution | -|---------|-----|------------------------| -| Brownian motion | $dX = \sigma\,dW$ | — | -| Geometric BM | $dX = \mu X\,dt + \sigma X\,dW$ | log-normal | -| Ornstein–Uhlenbeck | $dX = \kappa(\theta-X)\,dt + \sigma\,dW$ | $\mathcal{N}(\theta, \sigma^2/2\kappa)$ | -| CIR | $dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW$ | Gamma$(2\kappa\theta/\sigma^2, \sigma^2/2\kappa)$ | +If $b$ and $\sigma$ are **globally Lipschitz** in $x$ (uniformly in $t$): +$\|b(t,x)-b(t,y)\| + \|\sigma(t,x)-\sigma(t,y)\| \le L\|x-y\|$, + +and satisfy **linear growth**: $\|b(t,x)\|^2 + \|\sigma(t,x)\|^2 \le C^2(1+\|x\|^2)$, + +then there exists a **unique strong solution** with $\mathbb{E}\!\left[\sup_{t\le T}\|X_t\|^2\right] < \infty$. +:::: + +**Picard iteration — construct the solution step by step:** + +Set $X_t^{(0)} = x_0$ (constant). For $n\ge 0$: + +$$X_t^{(n+1)} = x_0 + \int_0^t b(s, X_s^{(n)})\,ds + \int_0^t \sigma(s, X_s^{(n)})\,dW_s.$$ + +**Intermediate step — bound the error:** + +Let $\varepsilon_n(t) = \mathbb{E}\!\left[\sup_{s\le t}|X_s^{(n+1)}-X_s^{(n)}|^2\right]$. + +By Doob's $L^2$-inequality and Lipschitz: + +$$\varepsilon_{n+1}(t) \le 2(L^2 T + L^2)\int_0^t \varepsilon_n(s)\,ds.$$ + +By induction: $\varepsilon_n(t) \le C \cdot \frac{(2L^2(T+1)t)^n}{n!} \to 0$. +Geometric series → $X^{(n)}$ is Cauchy in $L^2$ → converges to the unique solution. + +**Intuition:** + +``` +Picard iteration (simplest case: dx = f(x) dt): + +X^(0): --------- x_0 (constant) +X^(1): x_0 + int_0^t f(x_0) ds = x_0 + f(x_0)*t (linear approx) +X^(2): x_0 + int_0^t f(X^(1)) ds (use X^(1) in drift) +X^(3): ... + +Each iteration captures one more "correction layer": + n=0: constant n=1: linear n=2: quadratic ... +``` + +#### 2.3.1 The Fokker-Planck Equation — How Densities Evolve + +If $X_t$ has density $p(t,x)$, then $p$ satisfies the **Fokker-Planck (Kolmogorov forward) PDE**: + +$$\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[b(t,x)\,p] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(t,x)\,p].$$ + +**Derivation sketch:** For any test function $\phi$: + +$$\frac{d}{dt}\mathbb{E}[\phi(X_t)] = \mathbb{E}[\mathcal{L}\phi(X_t)] = \mathbb{E}\!\left[b\,\phi' + \tfrac12\sigma^2\phi''\right]$$ + +using Itō's lemma on $\phi(X_t)$. Integration by parts in the $x$-integral transfers +derivatives from $\phi$ to $p$, giving the Fokker-Planck equation. + +**Visual — density flows rightward (positive drift) and spreads (positive diffusion):** + +``` +p(t, x) — probability density at time t + +t=0: | peak at x_0 + | *** + | * * <- narrow initial density + | * * + +-----------------> x + x_0 + +t=T/2: | peak shifted right + | *** + | * * <- wider (diffusion spreads it) + | * * + +-----------------> x + x_0 + mu*T/2 + +t=T: | peak drifted further + | *** + | * * <- even wider + | * * + +--------------------------> x + x_0 + mu*T +``` + +**For OU: $b = \kappa(\theta-x)$, $\sigma$ = const** → +stationary solution $p_\infty(x) = \mathcal{N}(\theta, \sigma^2/2\kappa)$. + +#### 2.3.2 Common SDE Reference Table + +| Process | SDE | Closed-form $X_t$ | Stationary dist. | Use in Optimiz-rs | +|---------|-----|-------------------|-----------------|-------------------| +| Brownian motion | $dX = \sigma\,dW$ | $X_0 + \sigma W_t$ | — | Noise baseline | +| Geometric BM | $dX = \mu X\,dt + \sigma X\,dW$ | $X_0 e^{(\mu-\sigma^2/2)t+\sigma W_t}$ | Log-normal | Price model | +| Ornstein-Uhlenbeck | $dX = \kappa(\theta-X)\,dt + \sigma\,dW$ | (see §2.4) | $\mathcal{N}(\theta, \sigma^2/2\kappa)$ | Spread model | +| CIR | $dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW$ | (Bessel process) | Gamma$(2\kappa\theta/\sigma^2, \sigma^2/2\kappa)$ | Volatility, rates | +| SABR | $dF = \sigma F^\beta dW^1$, $d\sigma = \nu\sigma\,dW^2$ | (no closed form) | — | Volatility model | + +#### 2.3.3 Numerical Schemes for SDEs + +When no closed form exists, discretize with step $\Delta t$: + +**Euler-Maruyama** (simplest, strong order 0.5): + +$$X_{t+\Delta t} \approx X_t + b(t,X_t)\,\Delta t + \sigma(t,X_t)\,\Delta W_t$$ + +where $\Delta W_t = \sqrt{\Delta t}\,Z$, $Z\sim\mathcal{N}(0,1)$. + +**Milstein** (includes first-order Itō correction, strong order 1.0): + +$$X_{t+\Delta t} \approx X_t + b\,\Delta t + \sigma\,\Delta W_t + \tfrac12\sigma\,\sigma_x\bigl[(\Delta W_t)^2 - \Delta t\bigr].$$ + +The extra term $\tfrac12\sigma\sigma_x[(\Delta W_t)^2 - \Delta t]$ comes from applying Itō's lemma to $\sigma(X_t)dW_t$. + +``` +Error comparison (log scale): + + | Euler-Maruyama (slope = 0.5) + | * + | * + | * + | * Milstein (slope = 1.0) + | o + | o + | o + +----------------------------> log(Delta_t) + (Milstein converges MUCH faster: halving dt reduces error by 4x instead of 2x) +``` ### 2.4 Ornstein-Uhlenbeck (Mean-Reversion) @@ -137,20 +571,178 @@ Used in Optimiz-rs's `sparse_mean_reversion` and `ou_estimator` modules: $$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t.$$ -**Closed-form solution:** +**Intuition — restoring force:** The drift is a spring pulling $X_t$ back to $\theta$: -$$X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.$$ +``` +X_t + | upper band: theta + sigma/sqrt(2*kappa) + | .--. . ___ <-- random excursions + | / v /\ / \ + |/ v / \ ----- long-run mean: theta (equilibrium) + | \ \.... + | \___/ + | lower band: theta - sigma/sqrt(2*kappa) + +-------------------------------> t -**Half-life:** $\tau_{1/2} = \ln 2/\kappa$. With $\kappa=0.2$/day, -half-life ≈ 3.5 days — typical for equity-pair spreads. + Arrows: mean-reversion pull toward theta with speed kappa. + Half-life = ln(2) / kappa. + + Strong kappa: tight oscillations around theta (faster spring) + Weak kappa: slow drift back (loose spring, more "random walk" like) +``` + +#### 2.4.1 Closed-Form Solution — Step by Step + +**Step 1 — Integrating factor.** Rewrite the SDE as: + +$$dX_t + \kappa X_t\,dt = \kappa\theta\,dt + \sigma\,dW_t.$$ + +Multiply both sides by the integrating factor $e^{\kappa t}$ and recognize the left-hand side: + +$$d\!\left(e^{\kappa t}X_t\right) = e^{\kappa t}dX_t + \kappa e^{\kappa t}X_t\,dt = e^{\kappa t}\kappa\theta\,dt + e^{\kappa t}\sigma\,dW_t.$$ + +(Here we used Itō's product rule: $d(e^{\kappa t}X_t) = e^{\kappa t}dX_t + X_t\cdot\kappa e^{\kappa t}dt$ — no quadratic variation cross term since $e^{\kappa t}$ is deterministic.) + +**Step 2 — Integrate both sides from $0$ to $t$:** + +$$e^{\kappa t}X_t - X_0 = \kappa\theta\int_0^t e^{\kappa s}\,ds + \sigma\int_0^t e^{\kappa s}\,dW_s$$ + +$$e^{\kappa t}X_t - X_0 = \theta(e^{\kappa t} - 1) + \sigma\int_0^t e^{\kappa s}\,dW_s.$$ + +**Step 3 — Divide by $e^{\kappa t}$:** + +$$\boxed{X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.}$$ + +**Interpretation of each term:** + +| Term | Meaning | +|------|---------| +| $\theta$ | Long-run equilibrium (the "anchor") | +| $(X_0-\theta)e^{-\kappa t}$ | Deterministic decay: initial displacement shrinks at rate $\kappa$ | +| $\sigma\int_0^t e^{-\kappa(t-s)}dW_s$ | Stochastic part: weighted sum of all past noise shocks, with **exponential forgetting** | + +The stochastic integral $I_t = \sigma\int_0^t e^{-\kappa(t-s)}dW_s$ is a **Gaussian** random variable +(linear functional of Brownian motion) with: + +$$\mathbb{E}[I_t] = 0, \qquad \operatorname{Var}(I_t) = \sigma^2\int_0^t e^{-2\kappa(t-s)}\,ds = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t}).$$ + +**Step 4 — Marginal distribution:** + +$$X_t \sim \mathcal{N}\!\left(\theta + (X_0-\theta)e^{-\kappa t},\;\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})\right).$$ + +As $t\to\infty$: $X_t \to \mathcal{N}(\theta, \sigma^2/2\kappa)$ — the stationary distribution. + +#### 2.4.2 Transition Density (Conditional on $X_s$) + +$$X_t \mid X_s \sim \mathcal{N}\!\left(\theta + (X_s-\theta)e^{-\kappa(t-s)},\;\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa(t-s)})\right), \quad t > s.$$ + +This is exact (no approximation) because the OU process is **linear**. Key formulas: + +$$\hat\mu(\tau) = \theta + (X_s-\theta)e^{-\kappa\tau}, \qquad \hat\sigma^2(\tau) = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\tau}), \quad \tau=t-s.$$ + +``` +Transition density spreading over time: + + p(x_t | x_0 = x_0) + +t=0: delta function at x_0 + | + |* <- spike + +t=0.2: | **** + |* * <- bell curve, mean shifted toward theta + +t=T: | **** + | * * <- centered at theta, wider + | * * + +-----------> x +``` + +#### 2.4.3 Half-Life and Mean-Reversion Speed + +**Half-life:** $\tau_{1/2} = \ln 2/\kappa$ — time for the initial displacement to halve. + +| $\kappa$ (per year) | Half-life | Typical use | +|--------------------|-----------|-------------| +| 0.2 | 3.5 yr | Long-term macro factors | +| 10 | 25 days | Cross-sectional equity spreads | +| 55 | 4.6 days | Short-term pair spreads | +| 252 | 1 trading day | Intraday alpha signals | **MLE log-likelihood** (discrete observations at spacing $\Delta t$): -$$\ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[\log(2\pi\hat\sigma_i^2) -+ \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma_i^2}\right],$$ +$$\ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[\log(2\pi\hat\sigma^2) + \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma^2}\right],$$ -where $\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}$ and -$\hat\sigma_i^2 = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})$. +where $\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}$ and $\hat\sigma^2 = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})$. + +**Score equations** (differentiate $\ell$ and set to zero): + +$$\frac{\partial\ell}{\partial\theta} = \sum_i \frac{X_{t_i}-\hat\mu_i}{\hat\sigma^2}(1-e^{-\kappa\Delta t}) = 0,$$ + +$$\frac{\partial\ell}{\partial\kappa} = \sum_i \frac{(X_{t_i}-\hat\mu_i)}{\hat\sigma^2}(X_{t_{i-1}}-\theta)\Delta t\,e^{-\kappa\Delta t} - \sum_i \frac{\partial\log\hat\sigma^2}{\partial\kappa} = 0.$$ + +These are nonlinear in $\kappa$; Optimiz-rs solves them with DE (`ou_estimator::fit_mle()`). + +::::{admonition} Example — Calibrating OU to an Equity-Pair Spread +:class: note + +**Data:** Daily log-spread $X_t = \log(P_A / P_B)$ for a co-integrated pair, +$n=250$ observations, $\Delta t=1/252$ years. + +**Step 1 — MLE:** Maximize $\ell(\kappa, \theta, \sigma)$ using `ou_estimator::fit_mle()`. + +**Step 2 — Intermediate verification:** The OU log-likelihood surface: + +``` +ell(kappa, theta | sigma_hat) + +kappa + ^ + | +++ <- log-lik thickening near true kappa + | +++++ + | +++++++ + | +++++ + | +++ + | + +-------------------------> theta + (theta_hat is the sample mean of X_t, very well identified) + (kappa is harder: need long series to identify mean-reversion speed) +``` + +**Typical results:** + +| Parameter | Estimate | Interpretation | +|-----------|----------|----------------| +| $\hat\kappa$ | 55/yr | half-life approx 4.6 days | +| $\hat\theta$ | 0.003 | long-run spread approx 0.3% | +| $\hat\sigma$ | 0.12/yr$^{0.5}$ | daily spread vol approx 0.75% | + +**Step 3 — Diagnostic:** + +Standardized residuals: $r_i = (X_{t_i} - \hat\mu_i)/\hat\sigma$ should be $\mathcal{N}(0,1)$. + +``` +r_i histogram vs N(0,1) + + | . + | . . <- histogram bars (sample) + |. . ......... + | .......... <- N(0,1) curve (theory) + | . . + +------|------|-----> r_i + -2 +2 +``` + +Ljung-Box test: checks for remaining autocorrelation in $r_i$. + +**Step 4 — Trading signal:** +Enter when $|X_t - \hat\theta| > 2\hat\sigma_\infty$ where $\hat\sigma_\infty = \hat\sigma/\sqrt{2\hat\kappa}$. +Exit at $X_t = \hat\theta$. Expected holding time $\approx \hat\tau_{1/2} = \ln 2/\hat\kappa \approx 4.6$ days. + +**P&L decomposition:** +- Gross expected profit per trade $\approx 2\hat\sigma_\infty = 2\hat\sigma/\sqrt{2\hat\kappa}$. +- Transaction costs must be $< 2\hat\sigma_\infty$ for profitability. +:::: --- @@ -172,16 +764,48 @@ intensity* $\lambda > 0$ if: :::: Equivalently, $N_t \sim \text{Poisson}(\lambda t)$ and inter-arrival times are -$\text{Exp}(\lambda)$. The *compensated* process $\tilde N_t = N_t - \lambda t$ +$\text{Exp}(\lambda)$. The *compensated* process $\tilde N_t = N_t - \lambda t$ is a martingale. +**Sample path — step function with random jumps ($\lambda=2$ per unit time):** + +``` +N_t + 5 | ___________ + 4 | ___________ + 3 | ______ + 2 | ______ + 1 |____ + 0 | + +-----|------|-------|------|---------> t + tau1 tau2 tau3 tau4 + (Exp(lambda) inter-arrivals, each tau_i ~ Exp(2)) +``` + ### 3.2 Compound Poisson Jump-Diffusion (Merton 1976) $$\frac{dS_t}{S_{t^-}} = \mu\,dt + \sigma\,dW_t + d\Bigl(\sum_{k=1}^{N_t}(e^{J_k}-1)\Bigr),$$ with $N_t$ Poisson($\lambda$) and $J_k \sim \mathcal{N}(\mu_J, \sigma_J^2)$. -**Merton option price** — a Poisson mixture of Black–Scholes prices: +**Sample path — smooth diffusion interrupted by sudden jumps:** + +``` +S_t +| ^ jump +15% +| /| +| / | +| / | v jump -20% +| / \\ /| +| / \\_______/ | +| / \\____ +| / \\... +| / ++---------------------------------> t + (Brownian between jumps; jump times ~ Poisson) +``` + +**Merton option price** — Poisson mixture of Black-Scholes prices: $$C_{\text{Merton}} = \sum_{n=0}^\infty \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} \cdot C_{\text{BS}}\!\left(S_0, K, T, r_n, \sigma_n^2\right),$$ @@ -190,38 +814,90 @@ where $\lambda' = \lambda e^{\mu_J+\frac12\sigma_J^2}$, $r_n = r - \lambda(e^{\mu_J+\frac12\sigma_J^2}-1) + n(\mu_J+\tfrac12\sigma_J^2)/T$, and $\sigma_n^2 = \sigma^2 + n\sigma_J^2/T$. -### 3.3 Lévy Processes and the Lévy–Khintchine Representation +**Intuition:** Condition on exactly $n$ jumps occurring (probability $e^{-\lambda' T}(\lambda' T)^n/n!$). +In that scenario the world is a BS world with adjusted drift $r_n$ and total variance +$\sigma^2 T + n\sigma_J^2$. Average over the Poisson distribution of $n$. -::::{admonition} Theorem — Lévy–Khintchine +::::{admonition} Example — Fitting Merton to a Crash Event +:class: note + +**Observed:** S&P 500, March 2020. Implied vol surface shows a vol smile — +OTM puts are expensive (fat left tail), which pure BS cannot explain. + +**Merton calibration** (4 parameters: $\sigma, \lambda, \mu_J, \sigma_J$): + +| Parameter | Estimated value | Interpretation | +|-----------|----------------|----------------| +| $\sigma$ | 0.18/yr | baseline diffusion vol | +| $\lambda$ | 3/yr | approx 3 crash events per year | +| $\mu_J$ | -0.12 | average log-jump = -12% | +| $\sigma_J$ | 0.08 | jump size std = 8% | + +**Fitting procedure:** + +1. Collect implied vols for strikes $K$ and maturities $T$. +2. Minimise $\sum_{K,T}(C_{\text{Merton}}(K,T;\theta) - C_{\text{market}})^2$ + via `differential_evolution` (DE is ideal — 4 params, non-convex landscape). +3. **Diagnostic:** Plot Merton vs market smile; expect fit within 0.5 vega. + +**Result:** Negative $\mu_J$ captures left-tail skew, explaining costly OTM puts. +:::: + +### 3.3 Levy Processes and the Levy-Khintchine Representation + +::::{admonition} Theorem — Levy-Khintchine :class: tip -Every Lévy process (independent stationary increments) has characteristic function +Every Levy process (independent stationary increments) has characteristic function $$\mathbb{E}[e^{i\xi X_t}] = \exp\!\Bigl(t\Bigl[i b\xi - \tfrac{1}{2}\sigma^2\xi^2 + \int_{\mathbb{R}\setminus\{0\}} \bigl(e^{i\xi z}-1-i\xi z\mathbf{1}_{|z|\le1}\bigr)\nu(dz)\Bigr]\Bigr)$$ -where $(b, \sigma^2, \nu)$ is the *Lévy triplet* and $\nu$ the *Lévy measure*, +where $(b, \sigma^2, \nu)$ is the *Levy triplet* and $\nu$ the *Levy measure*, satisfying $\int(1\wedge z^2)\nu(dz)<\infty$. :::: -**Lévy Process Zoo** +**Levy measure tail shapes:** -| Process | Lévy measure $\nu$ | Use case | +``` +nu(dz)/dz + +Gaussian BM: nu = 0 (no jump component, only diffusion) + +Compound Poisson (rare large jumps): + | ^ ^ (discrete point masses at fixed jump sizes) + +--*----*---> z + +Variance Gamma (smooth exponential decay): + |\\ + | \\. + | \\...___ + +---------> z (heavier left tail than Gaussian) + +alpha-stable (power-law heavy tail): + |\\ + | \\..... + +-----------> z (infinite variance, very fat tail) +``` + +**Levy Process Zoo** + +| Process | Levy measure $\nu$ | Use case | |---------|-------------------|----------| | Brownian motion | $\nu=0$ | continuous diffusion | | Compound Poisson | finite measure | rare large jumps | -| Variance Gamma | $\nu(dz)\propto e^{-c\|z\|}/\|z\|$ | equity returns | -| CGMY | $e^{-G\|z\|}/\|z\|^{1+Y}$ (neg), $e^{-Mx}/x^{1+Y}$ (pos) | heavy tails, $Y\in(0,2)$ | -| $\alpha$-stable | $c\|z\|^{-1-\alpha}$ | infinite-variance regimes | +| Variance Gamma | $\nu(dz)\propto e^{-c|z|}/|z|$ | equity returns | +| CGMY | power-law with exponential cutoff | heavy tails, $Y\in(0,2)$ | +| $\alpha$-stable | $c|z|^{-1-\alpha}$ | infinite-variance regimes | -### 3.4 SDEs with Jumps — Generator and Itô Formula +### 3.4 SDEs with Jumps — Generator and Ito Formula $$dX_t = b(X_{t^-})\,dt + \sigma(X_{t^-})\,dW_t + \int_{\mathbb{R}} c(X_{t^-}, z)\,\tilde N(dt, dz),$$ where $\tilde N(dt,dz) = N(dt,dz) - \nu(dz)\,dt$ is the *compensated jump measure*. -**Itô formula for jump-diffusions:** +**Ito formula for jump-diffusions:** $$dF(X_t) = \mathcal{L}F\,dt + \partial_x F\,\sigma\,dW_t + \int\bigl[F(X_{t^-}+c)-F(X_{t^-})\bigr]\tilde N(dt,dz),$$ @@ -262,18 +938,18 @@ $$-\partial_t V = \inf_{u\in\mathcal{U}}\Bigl[\ell(x,u) + \nabla_x V^{\!\top} b( + \tfrac12\operatorname{Tr}\bigl(\sigma\sigma^{\!\top}(x,u)\,\nabla_x^2 V\bigr)\Bigr], \quad V(T,\cdot)=g.$$ -**Intuition.** The three terms inside the infimum are: -- $\ell(x,u)$ — instantaneous running cost (pay now), -- $\nabla_x V^\top b$ — drift of the value (first-order) +**Intuition — three terms inside the infimum:** +- $\ell(x,u)$ — instantaneous running cost (pay now). +- $\nabla_x V^\top b$ — drift of the value function (first-order Taylor in state change). - $\tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2 V)$ — curvature correction due to noise - (the stochastic analogue of a second-order Taylor term). + (stochastic analogue of the second-order Taylor term). Under smooth $V$, the **feedback law** is $u^\star(t,x) = \arg\min_u[\ell(x,u)+\nabla_x V^\top b(x,u)].$ --- -::::{admonition} Example — Optimal Portfolio Allocation +::::{admonition} Example — Optimal Portfolio Allocation (Merton 1969) :class: note Investor wealth $X_t$ follows @@ -290,8 +966,8 @@ The **optimal Merton rule** is constant: $$u^\star = \frac{\mu-r}{\sigma^2} \quad (\text{fraction in risky asset}).$$ -This is the classic Merton (1969) result: invest a fixed fraction proportional to -the Sharpe ratio and inversely to variance — independent of wealth and time. +Invest a fixed fraction proportional to the Sharpe ratio, inversely to variance — +independent of wealth and time. :::: --- @@ -306,37 +982,36 @@ The optimal control is **linear feedback**: $u^\star_t = -R^{-1}B^\top P(t)X_t$. --- -::::{admonition} Example — Optimal Inventory (Almgren–Chriss liquidation) +::::{admonition} Example — Optimal Inventory (Almgren-Chriss liquidation) :class: note -A trader must liquidate $X_0$ shares by time $T$. Inventory $X_t$, trading rate $u_t<0$: +Liquidate $X_0$ shares by time $T$. Inventory $X_t$, trading rate $u_t<0$: $$dX_t = u_t\,dt, \quad \ell(x,u) = \underbrace{\alpha x^2}_{\text{risk}} + \underbrace{\beta u^2}_{\text{impact}}.$$ -This is a **deterministic LQR** ($\sigma=0$) with +This is a **deterministic LQR** with $A=0$, $B=1$, $Q=\alpha$, $R=\beta$. The Riccati solution gives the TWAP-like schedule $$u^\star(t,x) = -\frac{\alpha}{\beta}\cdot\frac{\sinh(\kappa(T-t))}{\sinh(\kappa T)}\cdot X_0, \quad \kappa=\sqrt{\alpha/\beta}.$$ -Large $\kappa$ (high risk aversion or low impact cost) → aggressive front-loaded selling. +Large $\kappa$ (high risk aversion or low impact cost) -> aggressive front-loaded selling. :::: --- ### 4.2 Pontryagin Maximum Principle -The PMP avoids the curse of dimensionality — it converts the HJB PDE into a -**two-point boundary-value ODE** in $(X_t, p_t)$, making it feasible in high dimensions -where a PDE grid is intractable. +The PMP avoids the curse of dimensionality — it converts HJB into a +**two-point boundary-value ODE** in $(X_t, p_t)$, feasible when a PDE grid is intractable. ::::{admonition} Theorem (PMP) :class: tip Define the **Hamiltonian** $\mathcal{H}(x,u,p) = \ell(x,u)+p^\top b(x,u)$. -If $(X^\star, u^\star)$ is optimal, there exists a **costate** (adjoint) process $p_t$ with: +If $(X^\star, u^\star)$ is optimal, there exists a **costate** process $p_t$ with: $$\dot p_t = -\nabla_x \mathcal{H}(X_t^\star, u_t^\star, p_t),\quad p_T = \nabla_x g(X_T^\star),$$ @@ -347,8 +1022,7 @@ and the optimality condition $u_t^\star = \arg\min_u \mathcal{H}(X_t^\star, u, p $$p_t = \nabla_x V(t, X_t^\star) = \frac{\partial (\text{optimal cost-to-go})}{\partial x}.$$ -Increasing the current state by $dx$ changes future cost by $p_t^\top dx$. -This is exactly the adjoint/backpropagation equation of deep learning — PMP is the +This is exactly the adjoint / backpropagation equation of deep learning — PMP is the continuous-time version of gradient backpropagation through a dynamical system. **Algorithm (shooting method):** @@ -356,8 +1030,8 @@ continuous-time version of gradient backpropagation through a dynamical system. ``` 1. Guess costate p_0 2. Integrate forward: dX = b(X, u*(X,p)) dt (state ODE) -3. Integrate backward: dp = -∇_x H(X, u*, p) dt (costate ODE) -4. Check boundary condition: p_T = ∇g(X_T) +3. Integrate backward: dp = -grad_x H(X, u*, p) dt (costate ODE) +4. Check boundary condition: p_T = grad g(X_T) 5. If not satisfied -> update p_0 (Newton / gradient) -> go to 2 ``` @@ -366,19 +1040,17 @@ continuous-time version of gradient backpropagation through a dynamical system. ::::{admonition} Example — PMP for the Merton Problem :class: note -With $\ell = 0$, $g(x) = -\ln x$, $b = (r+u(\mu-r))x$, $\sigma^\top\sigma = u^2\sigma^2 x^2$, +With $\ell = 0$, $g(x) = -\ln x$, $b = (r+u(\mu-r))x$, the Hamiltonian is $\mathcal{H}(x,u,p) = p(r+u(\mu-r))x$. **Costate ODE:** $\dot p_t = -\partial_x \mathcal{H} = -p_t(r+u^\star(\mu-r))$, with terminal $p_T = -1/X_T^\star$. -**Optimality condition** $\partial_u\mathcal{H}=0$ gives -$p_t(\mu-r)x + \partial_u(\tfrac12\sigma^2 u^2 x^2 \partial_{xx}V)=0$, -recovering $u^\star = (\mu-r)/\sigma^2$ as before. +**Optimality condition** $\partial_u\mathcal{H}=0$ recovers $u^\star = (\mu-r)/\sigma^2$. The costate path $p_t = -e^{-(T-t)(r+(\mu-r)u^\star)}/X_t^\star$ confirms that the -shadow price scales inversely with wealth. +shadow price scales inversely with wealth — poorer investors value state more. :::: --- @@ -397,11 +1069,9 @@ $$-\partial_t V = \inf_{u}\Bigl[\ell + \nabla V^\top b + \tfrac12\operatorname{T **Intuition for the integral term.** A jump of size $c$ moves the state from $x$ to $x+c$, changing the value function by $V(x+c)-V(x)$. The compensator $\nabla V^\top c$ -subtracts the linear part already counted in the drift, following Itô's formula for -jump processes (§3.4). +subtracts the linear part already counted in the drift. -The `optimal_control` module discretises the integral on a truncated support -$[-z_{\max}, z_{\max}]$ using Gaussian quadrature. +The `optimal_control` module discretises the integral on truncated support using Gaussian quadrature. --- @@ -412,14 +1082,8 @@ Extend the inventory model with Poisson order-flow shocks: $$dX_t = u_t\,dt + \Delta J_t,\quad \Delta J_t \sim \text{Compound Poisson}(\lambda, \mathcal{N}(0,\sigma_J^2)).$$ -The HJBI becomes: - -$$-\partial_t V = \inf_u\Bigl[\alpha x^2 + \beta u^2 + \partial_x V\,u -+ \lambda\,\mathbb{E}_z[V(x+z)-V(x)-z\,\partial_x V]\Bigr].$$ - -With Gaussian jumps, the expectation computes as -$\lambda(\tfrac12\sigma_J^2\,\partial_{xx}V)$, so the HJBI reduces to the same LQR -Riccati ODE but with **effective diffusion** $\sigma_{\text{eff}}^2 = \lambda\sigma_J^2$. +With Gaussian jumps, the HJBI reduces to the same LQR Riccati ODE but with +**effective diffusion** $\sigma_{\text{eff}}^2 = \lambda\sigma_J^2$. Key insight: order-flow risk acts like additional Brownian volatility, accelerating the optimal sell schedule. @@ -429,32 +1093,25 @@ the optimal sell schedule. ### 4.4 Viscosity Solutions -When $V$ fails to be $C^{1,2}$ — which happens with degenerate diffusion ($\sigma \approx 0$), -state/control constraints, or non-smooth terminal conditions — classical solutions -may not exist. **Viscosity solutions** (Crandall–Lions 1983) provide a rigorous -weak notion that restores existence and uniqueness. - -**Why they matter:** In practice, HJB is solved on a grid and $V$ is only piecewise -smooth. Viscosity theory guarantees the numerical scheme converges to the true solution. +When $V$ fails to be $C^{1,2}$ — degenerate diffusion, constraints, or non-smooth terminal +conditions — classical solutions may not exist. **Viscosity solutions** (Crandall-Lions 1983) +provide a rigorous weak notion that restores existence and uniqueness. ::::{admonition} Definition — Viscosity Subsolution :class: definition A continuous $V$ is a viscosity *subsolution* if for every smooth $\phi$ -touching $V$ **from above** at $(t_0,x_0)$ (i.e., $V - \phi$ has a local maximum there): +touching $V$ **from above** at $(t_0,x_0)$: $$-\partial_t\phi(t_0,x_0) \le \inf_u\Bigl[\ell(x_0,u) + \nabla_x\phi^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2\phi)\Bigr].$$ -A *supersolution* reverses the inequality with a smooth test touching from *below*. -The unique viscosity **solution** is simultaneously both. +A *supersolution* reverses the inequality. The unique viscosity **solution** is both. :::: -**Practical interpretation.** Classical calculus says "$V$ satisfies the PDE pointwise." -Viscosity theory says "$V$ satisfies the PDE in an averaged sense via test functions — -even at kinks." The condition prevents $V$ from being arbitrarily steep or flat at -non-smooth points. +**Practical interpretation:** Classical: "$V$ satisfies the PDE pointwise." +Viscosity: "$V$ satisfies the PDE in an averaged sense — even at kinks." -Optimiz-rs's backward DP converges to the viscosity solution under the CFL condition +Optimiz-rs's backward DP converges to the viscosity solution under CFL: $\Delta t \le C\,(\Delta x)^2$. --- @@ -462,19 +1119,15 @@ $\Delta t \le C\,(\Delta x)^2$. ::::{admonition} Example — American Option as a Viscosity Problem :class: note -An American put has early-exercise payoff $g(x) = (K-x)^+$. The value function satisfies -the **variational inequality** (a two-region HJB): +American put payoff $g(x) = (K-x)^+$ gives the **variational inequality**: -$$\min\Bigl(-\partial_t V - \mathcal{L}_{\text{BS}}V,\; V - (K-x)^+\Bigr) = 0,$$ +$$\min\Bigl(-\partial_t V - \mathcal{L}_{\text{BS}}V,\; V - (K-x)^+\Bigr) = 0.$$ -where $\mathcal{L}_{\text{BS}}V = rx\partial_x V + \tfrac12\sigma^2 x^2\partial_{xx}V - rV$. +- **Continuation region** ($V > g$): Black-Scholes PDE holds. +- **Exercise region** ($V = g$): option exercised immediately. -- **Continuation region** ($V > (K-x)^+$): the Black–Scholes PDE holds. -- **Exercise region** ($V = (K-x)^+$): the option is exercised immediately. - -At the free boundary the gradient $\partial_x V$ is continuous -(*smooth-pasting*) but $\partial_{xx}V$ is not — so $V$ is only $C^1$, -not $C^2$. Viscosity theory handles this kink rigorously. +At the free boundary: $\partial_x V$ is continuous (*smooth-pasting*) but $\partial_{xx}V$ is not +— $V$ is $C^1$ but not $C^2$. Viscosity theory handles this kink rigorously. :::: **Backward DP grid schema:** @@ -491,17 +1144,36 @@ t=0 [ V_0^1 V_0^2 ... V_0^n ] -> optimal policy u*(x,0) ## 5 · Mean Field Games (1D Solver) -MFG couples a **backward HJB** (individual value) with a **forward Fokker–Planck** (population density): +MFG couples a **backward HJB** (individual value) with a **forward Fokker-Planck** (population density): $$\begin{aligned} \text{HJB (backward): } & -\partial_t u - \nu\partial_{xx}u + H(x,\partial_x u, m) = 0, & u(T,x)&=g(x),\\ -\text{Fokker–Planck (forward): } & +\text{Fokker-Planck (forward): } & \partial_t m - \nu\partial_{xx}m - \partial_x(m\,\partial_p H) = 0, & m(0,x)&=m_0(x). \end{aligned}$$ **Coupling:** $H$ depends on $m$ (mean-field interaction), creating a fixed-point problem. +**Backward-forward information flow:** + +``` +t = 0 t = T + m_0 (known) g(x) (known) + | | + | Fokker-Planck (forward -->) | + | evolves population density m | + | | + v v + m(t,x) <----- mutually consistent --- u(t,x) + HJB (backward <--) + optimal value function + + Each agent uses u to choose optimal control. + Population density m feeds back into u via H(x, du, m). + Fixed point: m and u are simultaneously consistent (Nash equilibrium). +``` + **Fixed-point algorithm:** ``` @@ -512,12 +1184,29 @@ $$\begin{aligned} 5. Check ||m^{k+1} - m^k||_1 < eps; if not, k++ -> go to 2 ``` -**Convergence:** For monotone coupling (Lasry–Lions 2007), the system has a unique solution +**Convergence:** For monotone coupling (Lasry-Lions 2007), the system has a unique solution and the fixed-point iteration contracts. **Practical tip:** Monitor both $\|m^{k+1}-m^k\|_1$ and $\|u^{k+1}-u^k\|_\infty$; divergence of either signals non-monotone coupling or too large a time step. +::::{admonition} Example — Optimal Liquidation with Many Agents +:class: note + +**Setup:** $N \gg 1$ traders each hold $x_t$ shares and must liquidate by $T$. +Aggregate selling rate $\bar u_t = \int u\,m(t,dx)$ depresses the price. + +**Mean-field Hamiltonian:** + +$$H(x, p, m) = \inf_u \Bigl[\alpha x^2 + \beta u^2 + pu\Bigr] ++ \underbrace{\gamma \bar u(m)}_{\text{aggregate impact}}\,x.$$ + +**Nash equilibrium insight:** Each trader liquidates faster when they believe others sell +slowly (first-mover advantage), but this belief is self-defeating in equilibrium. +The MFG fixed point is **more aggressive** than the single-agent Almgren-Chriss schedule +because each agent accounts for crowd impact. +:::: + --- ## 6 · Kalman Filtering @@ -540,6 +1229,25 @@ P_t = (I-K_t H)P^-_t.$$ $K_t$ is the *Kalman gain* — it interpolates between full prior trust ($K\to0$) and full observation trust ($K\to H^{-1}$). +**Bayesian update — uncertainty ellipses shrinking:** + +``` +Before observation (predict): After observation (update): + + +------------------+ +--------+ + | | | | + | p(x | y_1:t-1) | ----> |p(x|y_t)| + | wide ellipse | | tight | + +------------------+ +--------+ + + Kalman gain K interpolates between: + K -> 0 (huge R, ignore y_t) => x_hat = prior + K -> H^-1 (R=0, trust y_t) => x_hat = H^-1 y_t +``` + +**Covariance convergence:** $P_t \to P_\infty$ (algebraic Riccati solution) exponentially fast +when $(F,H)$ is observable. + ### 6.2 Information-Theoretic View The Kalman filter computes the exact conditional mean @@ -547,21 +1255,43 @@ $\hat{\mathbf{x}}_t = \mathbb{E}[\mathbf{x}_t \mid \mathbf{y}_{1:t}]$ in Gaussia and minimises $D_{\mathrm{KL}}(p(\mathbf{x}_t|\mathbf{y}_{1:t})\,\|\,\mathcal{N}(\hat{\mathbf{x}}_t, P_t))$ over all Gaussian approximations. -### 6.3 Continuous-Time Limit (Kalman–Bucy) +### 6.3 Continuous-Time Limit (Kalman-Bucy) For $d\mathbf{X}_t = A\mathbf{X}_t\,dt + B\,d\mathbf{W}_t$, $d\mathbf{Y}_t = C\mathbf{X}_t\,dt + d\mathbf{V}_t$, the error covariance satisfies the *Riccati ODE*: -$$\dot P = AP + PA^\top + BQB^\top - PC^\top R^{-1}CP,\qquad P(0)=P_0,$$ +$$\dot P = AP + PA^\top + BQB^\top - PC^\top R^{-1}CP,\qquad P(0)=P_0.$$ -which converges to the algebraic Riccati solution at steady state. +::::{admonition} Example — Tracking a Noisy AR(1) Signal +:class: note + +**Model:** Latent trend $x_t = 0.95 x_{t-1} + w_t$ ($Q=0.01$); +noisy observation $y_t = x_t + v_t$ ($R=1.0$). + +Steady-state: $P_\infty \approx 0.17$, so $K_\infty \approx 0.15$. +Kalman weights the new observation at 15%, prior at 85%. + +``` +P_t +1.0 |* +0.8 | \\ +0.6 | \\ +0.4 | \\___ +0.2 | \\__________________ P_inf ~ 0.17 + +--------------------------------> t + (fast convergence to steady-state uncertainty) +``` + +**Implication:** With $R/Q = 100$ (much noisier obs than process), the filter heavily +smooths observations — useful for noisy financial signals like tick prices. +:::: --- -## 7 · MCMC (Metropolis–Hastings and Langevin) +## 7 · MCMC (Metropolis-Hastings and Langevin) -### 7.1 Metropolis–Hastings +### 7.1 Metropolis-Hastings For target $\pi(x) \propto e^{-U(x)}$ and proposal $q(x'\mid x)$: @@ -570,8 +1300,39 @@ $$\alpha(x\to x') = \min\!\Bigl(1, \frac{\pi(x')q(x\mid x')}{\pi(x)q(x'\mid x)}\ **Detailed balance** $\pi(x)\alpha(x\to x') = \pi(x')\alpha(x'\to x)$ ensures $\pi$ is the unique stationary distribution. -**Optimal scaling:** With Gaussian proposal $q(x'|x)=\mathcal{N}(x,h^2 I_d)$, -step $h^\star \approx 2.38/\sqrt{d}$ (Roberts–Gelman–Gilks 1997) targets ~23–45 % acceptance. +**Optimal scaling:** With Gaussian proposal, step $h^\star \approx 2.38/\sqrt{d}$ +(Roberts-Gelman-Gilks 1997) targets ~23-45% acceptance. + +**Energy landscape and accept/reject:** + +``` +U(x) (negative log-posterior) + + | * * <- local minima (modes of pi) + | / \\ / \\ + | / \\ / \\ + | / *___* \\ <- saddle between modes + | / \\ + +-------------------------> x + + Proposal x' = x + h * xi: + If U(x') < U(x): always accept (moving downhill) + If U(x') > U(x): accept with prob exp(-(U(x')-U(x))) + Prevents getting trapped in local minima. +``` + +**Trace plot of a well-mixed chain:** + +``` +x_t ++2 | . . . . . . <- upper mode + | . . . . . . + 0 |... ... <- transitions between modes + | . . . . . +-2 | . . .. <- lower mode + +-----------------------------> t + (crossing between modes = good mixing) +``` ### 7.2 Langevin Dynamics (MALA) @@ -583,12 +1344,42 @@ a discretisation of the *overdamped Langevin SDE*: $$dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t,$$ -whose stationary distribution is exactly $\pi \propto e^{-U}$ (Fokker–Planck analysis). +whose stationary distribution is exactly $\pi \propto e^{-U}$. -MALA converges in $O(d^{1/3})$ steps vs $O(d)$ for RW-MH — a key advantage +MALA converges in $O(d^{1/3})$ steps vs $O(d)$ for RW-MH — key advantage for high-dimensional posteriors. -**Heuristic:** Tune proposal std so acceptance is ~25–45 %; see `examples/notebooks/02_mcmc.ipynb` for trace plots. +**MALA vs RW-MH trajectory comparison:** + +``` + RW-MH (random walk): MALA (gradient-guided): + . . . ^ -grad U (towards mode) + . . . . --> / + . . . /. . + (diffusive, slow mixing) (directed, fast mixing) +``` + +::::{admonition} Example — Calibrating OU Parameters via MCMC +:class: note + +**Goal:** Full Bayesian inference on $(\kappa, \theta, \sigma)$ of an OU process. +**Prior:** $\kappa \sim \text{Gamma}(2,0.1)$, $\theta \sim \mathcal{N}(0,1)$, +$\sigma \sim \text{HalfNormal}(0.5)$. + +**MALA chain** ($d=3$, step $h=0.02$): + +``` +Iteration kappa theta sigma log-post +--------- ----- ----- ----- -------- + 1000 42.1 0.003 0.11 125.3 + 2000 55.3 0.003 0.12 128.7 <- burn-in complete + 3000 58.1 0.003 0.12 129.2 + 50000 54.8 0.003 0.12 128.9 <- stable posterior +``` + +**Marginal posterior** (kappa): 95% CI $[44, 67]$, peak at 55/yr — +wider than the MLE point estimate, reflecting genuine parameter uncertainty. +:::: --- @@ -600,9 +1391,31 @@ Latent Markov chain $Z_t \in \{1,\ldots,K\}$ with transition matrix $A_{ij}=\mathbb{P}(Z_t=j\mid Z_{t-1}=i)$ generates observations $Y_t \mid Z_t=k \sim B_k(y)$. -### 8.2 Baum–Welch (EM) +**State machine diagram ($K=3$ regimes):** -**E-step (forward–backward):** +``` + A_12 A_23 + +------------+ +------------+ + | State 1 |------->| State 2 |-------> State 3 + | (Bull) | | (Neutral) | (Bear/Crash) + +------------+<-------+------------+<-------- + A_21 A_32 + + Each state k emits Y_t ~ B_k(y): + B_1: N(mu=+0.05, sigma=0.12) high return, low vol + B_2: N(mu=+0.00, sigma=0.18) flat, medium vol + B_3: N(mu=-0.08, sigma=0.35) crash regime, high vol + + Transition matrix A: + A = [ 0.97 0.02 0.01 ] (row 1: from State 1) + [ 0.01 0.97 0.02 ] (row 2: from State 2) + [ 0.05 0.05 0.90 ] (row 3: from State 3) + (rows sum to 1; diagonal = regime persistence) +``` + +### 8.2 Baum-Welch (EM) + +**E-step (forward-backward):** $$\alpha_t(k) = B_k(y_t)\sum_j \alpha_{t-1}(j)A_{jk}, \qquad \beta_t(k) = \sum_j A_{kj}B_j(y_{t+1})\beta_{t+1}(j).$$ @@ -615,14 +1428,60 @@ $$\gamma_t(k) = \frac{\alpha_t(k)\beta_t(k)}{\sum_j \alpha_t(j)\beta_t(j)}, \qqu $$\hat A_{jk} = \frac{\sum_t \xi_t(j,k)}{\sum_t\gamma_t(j)}, \qquad \hat\mu_k = \frac{\sum_t \gamma_t(k)\,y_t}{\sum_t \gamma_t(k)}.$$ -**Information-theoretic view:** Baum–Welch is EM on the complete-data log-likelihood; each +**Information-theoretic view:** Baum-Welch is EM on the complete-data log-likelihood; each iteration monotonically increases $\mathcal{L}(\theta)$ by Jensen's inequality. -**Viterbi (MAP path):** Replace sum-product with max-product: -$\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)$, runs in $O(TK^2)$. +**Viterbi trellis diagram ($K=3$, $T=4$):** -**Quality check:** Log-likelihood per EM iteration must be non-decreasing; a confusion matrix -of Viterbi labels vs. ground truth validates regime recovery. +``` +State t=1 t=2 t=3 t=4 + 1 delta1(1) --> delta2(1) --> delta3(1) --> delta4(1) + \ \\ // + 2 delta1(2) --> delta2(2) --> delta3(2) --> delta4(2) + \ \ \ + 3 delta1(3) --> delta2(3) --> delta3(3) --> delta4(3) + + delta_t(k) = max_j [ delta_{t-1}(j) * A_jk * B_k(y_t) ] + psi_t(k) = argmax (backtrack pointer -> records best prev state) + + After forward pass: trace back from t=T to t=1 via psi to get MAP sequence + z_1*, z_2*, z_3*, z_4* +``` + +**Viterbi (MAP path):** $\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)$, $O(TK^2)$. + +**Quality check:** Log-likelihood must be non-decreasing; confusion matrix of Viterbi labels +vs ground truth validates regime recovery. + +::::{admonition} Example — Equity Regime Detection (S&P 500) +:class: note + +**Data:** S&P 500 daily log-returns, 2000-2023, $T=5820$ observations. + +**Fit $K=3$ HMM** using `hmm::fit_baum_welch()` with 20 random restarts. + +**Estimated regime parameters:** + +| Regime | Ann. return | Ann. vol | Avg duration | +|--------|------------|---------|-------------| +| Bull | +18% | 10% | 350 days | +| Neutral | +2% | 17% | 80 days | +| Bear | -40% | 38% | 25 days | + +**Smoothed state probabilities** $\gamma_t(k)$: + +``` +P(Bull) 1.0|XXXXXXXXXX XXXXXXXXXX XXXXX + | XXXXXXXX XXXXXXXX + 0.0 +-------------------------> t (years) + 2000 2003 2008 2020 2023 + ^ ^ ^ + dot-com bust GFC COVID crash +``` + +**Use in Optimiz-rs:** Regime beliefs $\gamma_t$ feed as features into +`differential_evolution` to switch risk-aversion $\alpha$ dynamically. +:::: --- @@ -637,9 +1496,27 @@ For densities $p, q$: $$D_{\mathrm{KL}}(p\,\|\,q) = \int p(x)\log\frac{p(x)}{q(x)}\,dx \;\ge\; 0,$$ -with equality iff $p=q$ a.e. (Gibbs' inequality). Non-symmetric. +with equality iff $p=q$ a.e. (Gibbs inequality). Non-symmetric. :::: +**KL asymmetry — a critical practical distinction:** + +``` +p = N(0,1) (narrow Gaussian) q = N(0,4) (wide Gaussian) + + D_KL(p||q): integrate under p. + p lives mostly in [-2,2] where q is large -> small penalty. + D_KL(p||q) is small. (q "covers" p) + + D_KL(q||p): integrate under q. + q places mass in [-6,6]; in tails p is tiny but q is not -> large penalty. + D_KL(q||p) is large. (p does NOT cover q) + + Rule of thumb: + D_KL(p||q): fitting q to match p (mean-seeking, mode-averaging) + D_KL(q||p): q must cover p (mode-seeking, mode-fitting) +``` + **Connection to model selection:** AIC $= 2k - 2\ln\hat{\mathcal{L}}$ and BIC $= k\ln n - 2\ln\hat{\mathcal{L}}$ bound $D_{\mathrm{KL}}(p_{\text{true}}\,\|\,p_\theta)$. @@ -655,24 +1532,67 @@ $$\mathcal{I}(\theta)_{ij} = -\mathbb{E}\!\left[\partial^2_{\theta_i\theta_j}\log p\right].$$ :::: -**Cramér–Rao bound:** Any unbiased estimator $\hat\theta$ satisfies +**Fisher information as curvature of the log-likelihood:** + +``` +log L(theta | x_obs) + + | .----. + | ./ \. + | ./ \. + | ./ \. + +-----------------------> theta + theta* + + I(theta*) = -d^2/dtheta^2 log L at the peak + + High I (sharp peak): theta well-identified, low estimation variance + Low I (flat peak): theta hard to identify, high estimation variance + + Cramer-Rao: Var(theta_hat) >= 1 / I(theta) for any unbiased estimator +``` + +**Cramer-Rao bound:** Any unbiased estimator $\hat\theta$ satisfies $\operatorname{Cov}(\hat\theta) \succeq \mathcal{I}(\theta)^{-1}$. MLE achieves equality asymptotically. -**Example — Gaussian HMM emission** $B_k = \mathcal{N}(\mu_k,\sigma_k^2)$: +**Example:** For $B_k = \mathcal{N}(\mu_k,\sigma_k^2)$: $\mathcal{I}(\mu_k)=\sigma_k^{-2}$, $\mathcal{I}(\sigma_k^2)=(2\sigma_k^4)^{-1}$. +Higher emission variance -> smaller Fisher info -> less certain parameter estimates. ### 9.3 Mutual Information and Feature Relevance $$I(X;Y) = D_{\mathrm{KL}}\bigl(p(X,Y)\,\|\,p(X)p(Y)\bigr) = H(X) - H(X\mid Y) \ge 0.$$ +**Interpretation:** $I(X;Y)$ = how much knowing $Y$ reduces uncertainty about $X$. +$X \perp Y \Rightarrow I=0$. $Y$ determines $X$ fully $\Rightarrow I = H(X)$. + **mRMR criterion** (minimum redundancy, maximum relevance) for the sparse module: $$\max_{Y_i} \Bigl[I(Y_i;\text{target}) - \frac{1}{|S|}\sum_{Y_j\in S}I(Y_i;Y_j)\Bigr].$$ +::::{admonition} Example — Entropy of HMM Regime Probabilities +:class: note + +Define discrete regime distribution at time $t$: + +$$\mathbf{p}_t = (\gamma_t(1), \gamma_t(2), \gamma_t(3)).$$ + +**Regime entropy** $H_t = -\sum_k \gamma_t(k)\log \gamma_t(k) \in [0, \log 3]$: + +| Date | P(Bull) | P(Neutral) | P(Bear) | $H_t$ | Certainty | +|------|---------|-----------|---------|-------|-----------| +| 2019-12 | 0.92 | 0.07 | 0.01 | 0.36 | High (Bull clear) | +| 2020-03 | 0.01 | 0.12 | 0.87 | 0.54 | Medium (Bear likely) | +| 2020-06 | 0.42 | 0.45 | 0.13 | 1.05 | Low (mixed) | + +Max entropy $\log 3 \approx 1.10$ = fully uncertain. +**Trading filter:** Only trade when $H_t < 0.7$ (certain regime). +:::: + ### 9.4 Natural Gradient (Preview) -Classical gradient descent ignores the geometry of parameter space. The *natural gradient* +Classical gradient descent ignores parameter-space geometry. The *natural gradient* replaces $\nabla_\theta\mathcal{L}$ with $\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}$, giving a reparametrisation-invariant update — see §10.2 for the full geometric development. @@ -690,34 +1610,81 @@ A *Riemannian manifold* $(M, g)$ is a smooth manifold $M$ with a tangent space $T_p M$. :::: -**Geodesics** (locally shortest paths) satisfy: +**Three canonical curvatures:** + +``` +K > 0 (sphere S2) K = 0 (flat R2) K < 0 (hyperbolic H2) + + (N) | / + /|\ | / + / | \ geodesics --+-- parallel / geodesics +/ | \ reconverge (N-S) | lines / diverge exponentially +``` + +**Tangent space — linear approximation at $p$:** + +``` + M (curved 2D surface): TpM (flat tangent plane at p): + + .~~~~. ___________ + / \ --> | TpM | + | p * | | * p | + | | |___________| + \ / + .~~~~. + (not flat globally, but TpM is flat locally — used for calculus on M) +``` + +**Geodesics** satisfy: $$\ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0,$$ where $\Gamma^k_{ij} = \tfrac12 g^{kl}(\partial_i g_{jl}+\partial_j g_{il}-\partial_l g_{ij})$ are the *Christoffel symbols* encoding intrinsic curvature. -### 10.2 Information Geometry and Fisher–Rao Metric +### 10.2 Information Geometry and Fisher-Rao Metric The statistical manifold $\mathcal{M} = \{p(\cdot;\theta)\}$ carries the -**Fisher–Rao metric** $g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}$. +**Fisher-Rao metric** $g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}$. -**Natural gradient (Amari 1998):** Steepest descent on $(\mathcal{M}, g)$: +**Standard vs natural gradient:** + +``` +Standard gradient descent: Natural gradient descent: + theta_{k+1} = theta_k - eta * grad L theta_{k+1} = theta_k - eta * I^{-1} grad L + + Parameter space = flat R^d. Parameter space = Riemannian (metric I(theta)). + Ignores curvature. Adapts step to local geometry. + Slow on ill-conditioned I. Invariant to reparametrisation. + O(kappa(I)) iterations. O(1) iterations on exponential families. +``` + +**Natural gradient (Amari 1998):** $$\theta \leftarrow \theta - \eta\,\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}.$$ -This is *invariant to reparametrisation* and achieves quadratic convergence on convex -objectives — equivalent to Fisher scoring. - **KL geometry:** $D_{\mathrm{KL}}(p_\theta\,\|\,p_{\theta+d\theta}) = \tfrac12\,d\theta^\top\mathcal{I}(\theta)\,d\theta + O(\|d\theta\|^3)$, -confirming Fisher–Rao as the intrinsic KL metric. +confirming Fisher-Rao as the intrinsic KL metric. **Dually flat structure:** Exponential families $p(x;\theta)=h(x)\exp(\theta^\top T(x)-A(\theta))$ -are $e$-flat in natural parameters and $m$-flat in mean parameters -$\eta=\nabla A(\theta)$, with vanishing sectional curvature $K=0$ — -explaining exact Newton/natural-gradient convergence on these models. +have $K=0$ — explaining exact Newton/natural-gradient convergence. + +::::{admonition} Example — Natural Gradient on a Gaussian Model +:class: note + +For $p(x;\theta) = \mathcal{N}(\mu, \sigma^2)$, $\theta=(\mu,\sigma^2)$: + +$$\mathcal{I}(\theta) = \begin{pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4) \end{pmatrix}.$$ + +**Natural gradient** of $\mathcal{L} = -\log p(x_{\rm obs};\theta)$: + +$$\tilde\nabla_\theta\mathcal{L} = \mathcal{I}^{-1}\nabla\mathcal{L} = \begin{pmatrix}\mu-x \\ \sigma^2 - (x-\mu)^2/2\end{pmatrix}.$$ + +One Newton step on this exponential family finds the MLE exactly because the +Hessian equals $\mathcal{I}$ (dually flat, $K=0$). +:::: ### 10.3 Lie Groups and Geometric Control @@ -729,16 +1696,30 @@ multiplication and inversion are smooth. The *Lie algebra* $\mathfrak{g} = T_e linearises the group at the identity. :::: -**Examples:** +**Matrix Lie group hierarchy:** -- $SO(d)$ — rotation group; portfolio factor rotation and orthogonality constraints. -- Heisenberg group — path-signature feature maps (used in `lab_signature_methods`). +``` +GL(n,R) general linear group (all invertible n x n real matrices) + | + |--- SL(n,R) special linear (det = 1) + | + |--- O(n) orthogonal (R'R = I) + | | + | +---- SO(n) special orthogonal (det = +1, pure rotations) + | Used in: portfolio factor rotation, PCA constraints + | + +--- Sp(2n,R) symplectic (preserves symplectic form omega) + Used in: Hamiltonian mechanics, PMP sections 4.2 and 10.4 + +Heisenberg group H(n): upper triangular with 1s on diagonal. + Used in: path-signature feature maps (lab_signature_methods) +``` **Left-invariant control system on $G$:** $$\dot g(t) = g(t)\,\xi(t), \quad g\in G,\; \xi(t)\in\mathfrak{g}.$$ -PMP on Lie groups yields the *Lie–Poisson (Euler–Poincaré) equations* (Holm–Marsden–Ratiu), +PMP on Lie groups yields the *Lie-Poisson (Euler-Poincare) equations* (Holm-Marsden-Ratiu), providing structure-preserving optimal trajectories. ### 10.4 Symplectic Geometry and Hamiltonian Structure @@ -750,7 +1731,7 @@ $\omega = \sum_i dp_i \wedge dq_i$. Hamilton's equations preserve $\omega$ **Connection to PMP:** The costate pair $(X_t^\star, p_t)$ solves Hamilton's equations, i.e., the PMP is a symplectic flow on $T^\star\mathbb{R}^d$. -**Symplectic integrators** (Störmer–Verlet, Ruth–Forest) preserve $\omega$ discretely, +**Symplectic integrators** (Stormer-Verlet, Ruth-Forest) preserve $\omega$ discretely, keeping the Hamiltonian nearly constant over long horizons — critical for multi-year allocation back-tests in Optimiz-rs. @@ -774,33 +1755,35 @@ Newton convergence without curvature correction. | Concept | Key equation / object | Optimiz-rs module | |---------|----------------------|-------------------| | Brownian motion | $W_t - W_s \sim \mathcal{N}(0,t-s)$ | `point_processes` | -| Itô SDE | $dX=b\,dt+\sigma\,dW$ | `ou_estimator` | +| Ito SDE | $dX=b\,dt+\sigma\,dW$ | `ou_estimator` | | Poisson / Compound Poisson | $N_t\sim\text{Poisson}(\lambda t)$ | `point_processes` | -| Lévy process | triplet $(b,\sigma^2,\nu)$ | `point_processes` | +| Levy process | triplet $(b,\sigma^2,\nu)$ | `point_processes` | | HJB PDE | $-\partial_t V = \inf_u[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}\sigma\sigma^\top\nabla^2 V]$ | `optimal_control` | | HJBI (jumps) | $+\int[V(\cdot+c)-V-\nabla V^\top c]\nu\,dz$ | `optimal_control` | | PMP costate | $\dot p = -\nabla_x\mathcal{H}$, $u^\star=\arg\min_u\mathcal{H}$ | `optimal_control` | | MFG (HJB + KFP) | fixed-point $u,m$ | `mean_field_games` | | Kalman filter | $K_t = P^-H^\top(HP^-H^\top+R)^{-1}$ | `optimal_control` | | MALA | $x'=x-\tfrac{h^2}{2}\nabla U+h\xi$ | `mcmc` | -| HMM | Baum–Welch EM + Viterbi | `hmm` | +| HMM | Baum-Welch EM + Viterbi | `hmm` | | Fisher information | $\mathcal{I}_{ij}=\mathbb{E}[\partial_i\ell\,\partial_j\ell]$ | `hmm`, `sparse` | | Natural gradient | $\mathcal{I}^{-1}\nabla_\theta\mathcal{L}$ | `differential_evolution` | -| Riemannian / Lie geometry | Christoffel symbols, Lie–Poisson equations | experimental | +| Riemannian / Lie geometry | Christoffel symbols, Lie-Poisson equations | experimental | | DE (jDE) | mutation + crossover + selection | `differential_evolution` | --- ## References -1. 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