\documentclass[11pt,a4paper]{article} %% ── Encoding & Fonts ──────────────────────────────────────────────────────── \usepackage{fontspec} \setmainfont{Latin Modern Roman} \setsansfont{Latin Modern Sans} \setmonofont{Latin Modern Mono} %% ── Maths ─────────────────────────────────────────────────────────────────── \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{bm} % \bm for bold math %% ── Layout & Typography ───────────────────────────────────────────────────── \usepackage[a4paper, margin=2.5cm]{geometry} \usepackage{microtype} \usepackage{parskip} \usepackage{setspace} \setstretch{1.1} %% ── Colour ────────────────────────────────────────────────────────────────── \usepackage{xcolor} \definecolor{optblue}{HTML}{1A3A5C} \definecolor{optgold}{HTML}{C8960C} \definecolor{defbg}{HTML}{EEF4FB} \definecolor{thmbg}{HTML}{FFF8EC} \definecolor{codebg}{HTML}{F5F5F5} \definecolor{notebg}{HTML}{EAFAF1} \definecolor{optgreen}{HTML}{1A7A4A} \definecolor{warnbg}{HTML}{FEF3CD} \definecolor{optred}{HTML}{C0392B} %% ── Theorem environments ───────────────────────────────────────────────────── \usepackage{tcolorbox} \tcbuselibrary{skins,breakable} \newtcolorbox{defbox}[1]{ breakable, enhanced, colback=defbg, colframe=optblue, coltitle=white, fonttitle=\bfseries\sffamily, title={Definition — #1}, arc=3pt, boxrule=0.6pt, left=6pt, right=6pt, top=4pt, bottom=4pt } \newtcolorbox{thmbox}[1]{ breakable, enhanced, colback=thmbg, colframe=optgold, coltitle=white, fonttitle=\bfseries\sffamily, title={Theorem — #1}, arc=3pt, boxrule=0.6pt, left=6pt, right=6pt, top=4pt, bottom=4pt } \newtcolorbox{codebox}{ breakable, enhanced, colback=codebg, colframe=gray!40, arc=2pt, boxrule=0.4pt, left=8pt, right=8pt, top=4pt, bottom=4pt, fontupper=\ttfamily\small } %% ── Tables ─────────────────────────────────────────────────────────────────── \usepackage{booktabs} \usepackage{tabularx} \usepackage{array} \newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}} \newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}} %% ── Lists ──────────────────────────────────────────────────────────────────── \usepackage{enumitem} \setlist[itemize]{itemsep=2pt, topsep=4pt} \setlist[enumerate]{itemsep=2pt, topsep=4pt} %% ── Section styling ────────────────────────────────────────────────────────── \usepackage{titlesec} \titleformat{\section}{\large\bfseries\sffamily\color{optblue}}{\thesection}{0.7em}{}[\color{optblue}\titlerule] \titleformat{\subsection}{\normalsize\bfseries\sffamily\color{optblue}}{\thesubsection}{0.6em}{} \titleformat{\subsubsection}{\normalsize\itshape\sffamily}{\thesubsubsection}{0.5em}{} %% ── Header / Footer ────────────────────────────────────────────────────────── \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \fancyhead[L]{\sffamily\small\color{gray} Optimiz-rs — Mathematical Foundations} \fancyhead[R]{\sffamily\small\color{gray} \today} \fancyfoot[C]{\sffamily\small\color{gray}\thepage} \renewcommand{\headrulewidth}{0.4pt} %% ── Hyperlinks ─────────────────────────────────────────────────────────────── \usepackage[unicode, colorlinks=true, linkcolor=optblue, urlcolor=optblue, citecolor=optgold]{hyperref} \usepackage{bookmark} %% ── Listings (pseudocode) ──────────────────────────────────────────────────── \usepackage{listings} \lstset{ backgroundcolor=\color{codebg}, basicstyle=\ttfamily\small, breaklines=true, frame=single, framerule=0.4pt, rulecolor=\color{gray!40}, xleftmargin=8pt, xrightmargin=8pt, aboveskip=6pt, belowskip=6pt } %% ── Misc macros ────────────────────────────────────────────────────────────── \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\PP}{\mathbb{P}} \newcommand{\KL}[2]{D_{\mathrm{KL}}\!\left(#1\,\|\,#2\right)} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\N}{\mathcal{N}} %% ───────────────────────────────────────────────────────────────────────────── \begin{document} \section{Mathematical Foundations}\label{mathematical-foundations} 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/}. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \subsection{1 · Differential Evolution (DE)}\label{differential-evolution-de} \subsubsection{Background}\label{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 \(\{\mathbf{x}_{i,g}\} \subset \mathbb{R}^d\). \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} \begin{verbatim} 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 \end{verbatim} \begin{itemize} \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} \subsubsection{1.2 Operators}\label{operators} {\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: \[dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t.\] \textbf{Intuition --- restoring force:} The drift is a spring pulling \(X_t\) back to \(\theta\): \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 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. \subsubsection{3.1 Poisson Process}\label{poisson-process} \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: 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} 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{Sample path --- step function with random jumps (\(\lambda=2\) per unit time):} \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} \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{tcolorbox} \textbf{Levy measure tail shapes:} \begin{verbatim} 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) \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:} \[dF(X_t) = \mathcal{L}F\,dt + \partial_x F\,\sigma\,dW_t + \int\bigl[F(X_{t^-}+c)-F(X_{t^-})\bigr]\tilde N(dt,dz),\] where the \emph{generator} is \[\mathcal{L}F = b\,\partial_x F + \tfrac12\sigma^2\partial_{xx}F + \int\bigl[F(x+c)-F(x)-c\,\partial_x F\bigr]\nu(dz).\] \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \subsection{4 · Optimal Control (HJB, PMP, Jumps)}\label{optimal-control-hjb-pmp-jumps} \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 \[dX_t = b(X_t,u_t)\,dt + \sigma(X_t,u_t)\,dW_t,\] and we minimise the total expected cost \[J(t,x;u) = \mathbb{E}\!\left[\int_t^T \ell(X_s,u_s)\,ds + g(X_T)\,\Big|\,X_t=x\right].\] The \textbf{value function} \(V(t,x) = \inf_u J(t,x;u)\) satisfies: \[-\partial_t V = \inf_{u\in\mathcal{U}}\Bigl[\ell(x,u) + \nabla_x V^{\!\top} b(x,u) + \tfrac12\operatorname{Tr}\bigl(\sigma\sigma^{\!\top}(x,u)\,\nabla_x^2 V\bigr)\Bigr], \quad V(T,\cdot)=g.\] \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 $dX_t = (r + u_t(\mu-r))X_t\,dt + u_t\sigma X_t\,dW_t$, where $u_t\in\mathbb{R}$ is the fraction invested in the risky asset. 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} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \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} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \subsubsection{4.2 Pontryagin Maximum Principle}\label{pontryagin-maximum-principle} 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. \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: $$\dot p_t = -\nabla_x \mathcal{H}(X_t^\star, u_t^\star, p_t),\quad p_T = \nabla_x g(X_T^\star),$$ 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 condition: p_T = grad g(X_T) 5. If not satisfied -> update p_0 (Newton / gradient) -> go to 2 \end{verbatim} \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} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} 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). \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \subsubsection{4.3 HJB with Jumps (HJBI)}\label{hjb-with-jumps-hjbi} 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: $$dX_t = u_t\,dt + \Delta J_t,\quad \Delta J_t \sim \text{Compound Poisson}(\lambda, \mathcal{N}(0,\sigma_J^2)).$$ 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} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \subsubsection{4.4 Viscosity Solutions}\label{viscosity-solutions} 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{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 $(t_0,x_0)$: $$-\partial_t\phi(t_0,x_0) \le \inf_u\Bigl[\ell(x_0,u) + \nabla_x\phi^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2\phi)\Bigr].$$ A *supersolution* reverses the inequality. The unique viscosity **solution** is both. \end{tcolorbox} \textbf{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 CFL: \(\Delta t \le C\,(\Delta x)^2\). \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 — American Option as a Viscosity Problem},breakable] American put payoff $g(x) = (K-x)^+$ gives the **variational inequality**: $$\min\Bigl(-\partial_t V - \mathcal{L}_{\text{BS}}V,\; V - (K-x)^+\Bigr) = 0.$$ - **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{verbatim} 1. Initialise m^0 = m_0 (e.g. Gaussian) 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{verbatim} \textbf{Convergence:} For monotone coupling (Lasry-Lions 2007), the system has a unique solution and the fixed-point iteration contracts. \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. \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. **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:} \[\hat{\mathbf{x}}^-_t = F\hat{\mathbf{x}}_{t-1},\quad P^-_t = FP_{t-1}F^\top+Q.\] \textbf{Update:} \[K_t = P^-_t H^\top(HP^-_t H^\top + R)^{-1},\quad \hat{\mathbf{x}}_t = \hat{\mathbf{x}}^-_t + K_t(\mathbf{y}_t - H\hat{\mathbf{x}}^-_t),\quad P_t = (I-K_t H)P^-_t.\] \(K_t\) is the \emph{Kalman gain} --- it interpolates between full prior trust (\(K\to0\)) and full observation trust (\(K\to H^{-1}\)). \textbf{Bayesian update --- uncertainty ellipses shrinking:} \begin{verbatim} 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 \end{verbatim} \textbf{Covariance convergence:} \(P_t \to P_\infty\) (algebraic Riccati solution) exponentially fast when \((F,H)\) is observable. \subsubsection{6.2 Information-Theoretic View}\label{information-theoretic-view} 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. \subsubsection{6.3 Continuous-Time Limit (Kalman-Bucy)}\label{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 \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: \[x' = x - \tfrac{h^2}{2}\nabla U(x) + h\,\xi, \quad \xi\sim\mathcal{N}(0,I_d),\] a discretisation of the \emph{overdamped Langevin SDE}: \[dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t,\] whose stationary distribution is exactly \(\pi \propto e^{-U}\). MALA converges in \(O(d^{1/3})\) steps vs \(O(d)\) for RW-MH --- key advantage for high-dimensional posteriors. \textbf{MALA vs RW-MH trajectory comparison:} \begin{verbatim} RW-MH (random walk): MALA (gradient-guided): . . . ^ -grad U (towards mode) . . . . --> / . . . /. . (diffusive, slow mixing) (directed, fast mixing) \end{verbatim} \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:} \[\hat A_{jk} = \frac{\sum_t \xi_t(j,k)}{\sum_t\gamma_t(j)}, \qquad \hat\mu_k = \frac{\sum_t \gamma_t(k)\,y_t}{\sum_t \gamma_t(k)}.\] \textbf{Information-theoretic view:} Baum-Welch is EM on the complete-data log-likelihood; each iteration monotonically increases \(\mathcal{L}(\theta)\) by Jensen's inequality. \textbf{Viterbi trellis diagram (\(K=3\), \(T=4\)):} \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) 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 $p, q$: $$D_{\mathrm{KL}}(p\,\|\,q) = \int p(x)\log\frac{p(x)}{q(x)}\,dx \;\ge\; 0,$$ with equality iff $p=q$ a.e. (Gibbs inequality). Non-symmetric. \end{tcolorbox} \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 $p(x;\theta)$: $$\mathcal{I}(\theta)_{ij} = \mathbb{E}_{x\sim p}\!\left[\partial_{\theta_i}\log p\;\partial_{\theta_j}\log p\right] = -\mathbb{E}\!\left[\partial^2_{\theta_i\theta_j}\log p\right].$$ \end{tcolorbox} \textbf{Fisher information as curvature of the log-likelihood:} \begin{verbatim} 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 \end{verbatim} \textbf{Cramer-Rao bound:} Any unbiased estimator \(\hat\theta\) satisfies \(\operatorname{Cov}(\hat\theta) \succeq \mathcal{I}(\theta)^{-1}\). MLE achieves equality asymptotically. \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. \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{tcolorbox} \textbf{Three canonical curvatures:} \begin{verbatim} K > 0 (sphere S2) K = 0 (flat R2) K < 0 (hyperbolic H2) (N) | / /|\ | / / | \ geodesics --+-- parallel / geodesics / | \ reconverge (N-S) | lines / diverge exponentially \end{verbatim} \textbf{Tangent space --- linear approximation at \(p\):} \begin{verbatim} 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) \end{verbatim} \textbf{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 \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{tcolorbox} \textbf{Matrix Lie group hierarchy:} \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 Heisenberg group H(n): upper triangular with 1s on diagonal. Used in: path-signature feature maps (lab_signature_methods) \end{verbatim} \textbf{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 \emph{Lie-Poisson (Euler-Poincare) equations} (Holm-Marsden-Ratiu), providing structure-preserving optimal trajectories. \subsubsection{10.4 Symplectic Geometry and Hamiltonian Structure}\label{symplectic-geometry-and-hamiltonian-structure} 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). \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 \(T^\star\mathbb{R}^d\). \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. \subsubsection{10.5 Sectional Curvature and Landscape Geometry}\label{sectional-curvature-and-landscape-geometry} The sectional curvature \(K(\sigma)\) governs how quickly nearby geodesics diverge: \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} For exponential families in natural/mean parameters \(K=0\) --- explaining exact Newton convergence without curvature correction. \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}