docs(theory): enrich mathematical foundations with 10 sections
Sections added/expanded: - DE: operator table, jDE self-adaptive rules, convergence theorem - Stochastic Processes: 4-axiom BM, Ito calculus+lemma, SDE table, OU MLE - Jump Processes: Poisson, Merton formula, Levy-Khintchine theorem+table, SDE generator - Optimal Control: HJB boxed PDE, LQR Riccati, PMP costate theorem, HJBI jumps, viscosity solutions - Mean Field Games: HJB+KFP system, fixed-point pseudocode, Lasry-Lions convergence - Kalman Filtering: predict/update recursion, KL-minimization view, Kalman-Bucy Riccati - MCMC: MH accept-reject, optimal step h~2.38/sqrt(d), MALA+Langevin SDE - HMM: full Baum-Welch E/M (alpha/beta/gamma/xi), Viterbi max-product - Information Theory: Shannon entropy, KL+Gibbs, Fisher info+Cramer-Rao, mRMR, natural gradient preview - Differential Geometry: Riemannian manifold+geodesic+Christoffel, Fisher-Rao metric, Amari natural gradient, dually-flat exp families, Lie groups, symplectic geometry+PMP, sectional curvature - Quick Reference table (15 rows), 11 full references - Sphinx build clean (1 unrelated chardet warning only)
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# Mathematical Foundations
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This page collects the core equations driving Optimiz-rs’s Rust kernels, plus short intuition blurbs and micro-checks you can run in a notebook. For visuals and full walkthroughs, see the example notebooks in `examples/notebooks/`.
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This page develops the core mathematics underlying Optimiz-rs's Rust kernels — from first
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principles through advanced theory. Each section opens with a **definition block**, builds
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intuition through **examples**, and closes with a **notebook micro-check**. For complete
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walkthroughs see ``examples/notebooks/``.
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## Differential Evolution (DE)
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---
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We minimize $f: \mathbb{R}^d \to \mathbb{R}$ with a population $\{\mathbf{x}_{i,g}\}_{i=1}^N$.
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## 1 · Differential Evolution (DE)
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**Mutation (rand/1):**
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$$
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\mathbf{v}_{i,g} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g}),\quad r_1 \neq r_2 \neq r_3 \neq i.
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$$
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### Background
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**Intuition:** The differential term is a directional finite-difference estimate of the gradient; scaling $F$ sets the step length. Population diversity controls exploration.
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DE is a gradient-free population-based optimizer for :math:`f: \mathbb{R}^d \to \mathbb{R}`,
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not required to be smooth or convex. At generation :math:`g` we maintain :math:`N` candidate
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solutions :math:`\{\mathbf{x}_{i,g}\} \subset \mathbb{R}^d`.
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**Crossover (binomial):**
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$$
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u_{i,j,g} = \begin{cases}
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v_{i,j,g} & \text{if } \mathrm{Uniform}(0,1) < CR \text{ or } j = j_{\mathrm{rand}},\\
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x_{i,j,g} & \text{otherwise.}
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\end{cases}
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$$
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**Key insight:** The difference vector :math:`\mathbf{x}_{r_2}-\mathbf{x}_{r_3}` is an
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unbiased directional finite-difference of :math:`f`, so DE implicitly estimates curvature
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without Jacobians.
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**Selection (greedy):**
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$$
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\mathbf{x}_{i,g+1} = \begin{cases}
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\mathbf{u}_{i,g} & \text{if } f(\mathbf{u}_{i,g}) \le f(\mathbf{x}_{i,g}),\\
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\mathbf{x}_{i,g} & \text{otherwise.}
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\end{cases}
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$$
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### Operators
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**Self-adaptive jDE (used by Optimiz-rs):**
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$$
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F_i^{g+1} = \begin{cases}
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F_{\min} + r_1 \cdot F_{\max} & r_2 < \tau_1,\\
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F_i^{g} & \text{otherwise,}
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\end{cases}
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\qquad
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CR_i^{g+1} = \begin{cases}
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\mathrm{Uniform}(0,1) & r_3 < \tau_2,\\
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CR_i^{g} & \text{otherwise.}
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\end{cases}
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$$
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Typical $\tau_1, \tau_2 = 0.1$. This adaptation reduces manual tuning and improves robustness on multimodal landscapes.
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.. list-table::
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:header-rows: 1
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:widths: 20 50 30
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**Notebook check:** In `05_performance_benchmarks.ipynb`, plot $F_i$ and $CR_i$ histograms every 50 generations to verify adaptation is active (expect spread around 0.5–0.9 for $CR$ and 0.5–0.9 for $F$ on hard landscapes).
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* - Step
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- Formula
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- Role
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* - Mutation (rand/1)
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- :math:`\mathbf{v}_{i,g} = \mathbf{x}_{r_1} + F(\mathbf{x}_{r_2}-\mathbf{x}_{r_3})`
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- explore
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* - Binomial crossover
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- :math:`u_{i,j} = v_{i,j}` if :math:`U(0,1)<CR` or :math:`j=j_\text{rand}`
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- mix dimensions
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* - Greedy selection
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- :math:`\mathbf{x}_{i,g+1} = \mathbf{u}_{i,g}` iff :math:`f(\mathbf{u})\le f(\mathbf{x})`
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- exploit
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## Optimal Control (HJB)
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**Convergence (informal):** Under bounded population diversity and Lipschitz :math:`f`, the
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best-so-far value converges a.s. to a stationary point as :math:`N,g\to\infty` (Price et al. 2005).
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For dynamics $dX_t = b(X_t, u_t)\,dt + \sigma(X_t,u_t)\,dW_t$ with running cost $\ell$ and terminal cost $g$, the value function satisfies the Hamilton–Jacobi–Bellman PDE:
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$$
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-\partial_t V(t,x) = \inf_{u\in\mathcal{U}} \Big[ \ell(x,u) + \nabla_x V(t,x)^{\top} b(x,u) + \tfrac12 \operatorname{Tr}\big(\sigma\sigma^{\top}(x,u) \, \nabla_x^2 V(t,x)\big) \Big],\quad V(T,x) = g(x).
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$$
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### Self-Adaptive jDE (Optimiz-rs default)
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Optimiz-rs uses finite differences with backward time-stepping and optional policy iteration. On a uniform grid $(t_n, x_j)$:
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$$
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V^{n} = \min_{u}\Big\{ \ell(x_j,u)\,\Delta t + V^{n+1} + \nabla_x V^{n+1}\cdot b\,\Delta t + \tfrac12 \operatorname{Tr}(\sigma\sigma^{\top}\nabla_x^2 V^{n+1})\,\Delta t \Big\}.
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$$
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The control that attains the minimum yields the feedback policy $u^{\star}(x_j, t_n)$ exported by `compute_policy`.
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Parameters :math:`F,CR` are per-individual and reset stochastically each generation:
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**Interpretation:** HJB is dynamic programming in continuous time; $V$ encodes the optimal cost-to-go. The quadratic example in `03_optimal_control_tutorial.ipynb` shows $V$ becoming steeper where volatility is high or costs penalize deviation.
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.. math::
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## Mean Field Games (1D solver)
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F_i^{g+1} = \begin{cases} F_{\min} + r_1 F_{\max} & r_2 < \tau_1,\\ F_i^g & \text{otherwise,}\end{cases}
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\qquad
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CR_i^{g+1} = \begin{cases} U(0,1) & r_3 < \tau_2,\\ CR_i^g & \text{otherwise.}\end{cases}
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Optimiz-rs’s MFG module solves the coupled system for value $u$ and density $m$:
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$$
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\begin{aligned}
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-\partial_t u(t,x) - \nu\,\partial_{xx} u(t,x) + H\big(x,\partial_x u(t,x), m(t,x)\big) &= 0,\\
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\partial_t m(t,x) - \nu\,\partial_{xx} m(t,x) - \operatorname{div}\big(m(t,x) \, \partial_p H(x,\partial_x u, m)\big) &= 0,\\
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u(T,x) &= g(x), \qquad m(0,x) = m_0(x).
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\end{aligned}
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$$
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We use fixed-point iterations on the transport term with implicit diffusion (stable for $\nu > 0$) and normalize $m$ after each step to preserve mass.
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:math:`\tau_1=\tau_2=0.1` by default. On rugged landscapes this produces bimodal :math:`F`
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histograms concentrated near 0.8 — a sign the landscape is highly multimodal.
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**Practical tip:** Monitor $\|m^{k+1}-m^{k}\|_1$ and $\|u^{k+1}-u^{k}\|_\infty$; both appear in the notebook to diagnose non-convergence.
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**Notebook check** (``05_performance_benchmarks.ipynb``): Plot :math:`F_i, CR_i` histograms
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every 50 generations; expect values clustering in :math:`[0.5,0.9]` on hard problems.
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## Kalman Filtering
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---
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For linear-Gaussian state space models
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$$
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\begin{aligned}
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\mathbf{x}_{t} &= F\,\mathbf{x}_{t-1} + \mathbf{w}_{t}, && \mathbf{w}_t \sim \mathcal{N}(0, Q),\\
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\mathbf{y}_{t} &= H\,\mathbf{x}_{t} + \mathbf{v}_{t}, && \mathbf{v}_t \sim \mathcal{N}(0, R),
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\end{aligned}
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$$
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prediction and update follow:
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$$
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\begin{aligned}
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ext{Predict: } & \hat{\mathbf{x}}^-_t = F \hat{\mathbf{x}}_{t-1}, && P^-_t = F P_{t-1} F^{\top} + Q,\\
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ext{Update: } & K_t = P^-_t H^{\top} (H P^-_t H^{\top} + R)^{-1},\\
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& \hat{\mathbf{x}}_t = \hat{\mathbf{x}}^-_t + K_t(\mathbf{y}_t - H \hat{\mathbf{x}}^-_t),\\
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& P_t = (I - K_t H) P^-_t.
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\end{aligned}
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$$
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These steps back the `init_kalman_filter`, `kalman_predict`, and `kalman_update` helpers.
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## 2 · Stochastic Processes
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## MCMC (Metropolis–Hastings)
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These form the probabilistic backbone of all continuous-time models in Optimiz-rs.
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For target density $\pi(x)$ and proposal $q(x'\mid x)$:
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$$
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\alpha(x \to x') = \min\Big(1, \frac{\pi(x')\, q(x \mid x')}{\pi(x)\, q(x' \mid x)}\Big).
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$$
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Optimiz-rs uses symmetric Gaussian proposals (so $q$ cancels) by default, with optional bounds projection and burn-in.
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### 2.1 Brownian Motion
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**Heuristic:** Tune proposal std so acceptance is ~0.25–0.35 for moderate dimensions; see `examples/notebooks/02_mcmc.ipynb` for trace plots.
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.. admonition:: Definition — Wiener Process
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## Hidden Markov Models (HMM)
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A stochastic process :math:`W = (W_t)_{t\ge 0}` on :math:`(\Omega,\mathcal{F},\mathbb{P})`
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is a *standard Brownian motion* if:
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We maximize the likelihood of observations $\mathbf{y}$ under latent states $\mathbf{z}$ using Baum–Welch (EM):
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$$
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\mathcal{L}(\theta) = \sum_{t} \log \Big( \sum_{z_t} p(y_t \mid z_t, \theta) p(z_t \mid z_{t-1}, \theta) \Big).
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$$
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Forward–backward computes posteriors, then M-step re-estimates transition and emission parameters; Viterbi gives the MAP state path.
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1. :math:`W_0 = 0` a.s.
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2. Increments are **independent**: :math:`W_t - W_s \perp \mathcal{F}_s` for :math:`t>s`.
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3. :math:`W_t - W_s \sim \mathcal{N}(0, t-s)` for all :math:`0\le s<t`.
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4. Paths :math:`t\mapsto W_t(\omega)` are **continuous** a.s.
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**Quality check:** Plot log-likelihood per iteration; it should be non-decreasing. The HMM tutorial notebook includes a simple convergence plot and a confusion matrix for decoded states.
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**Key properties:**
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- :math:`\mathbb{E}[W_t] = 0`, :math:`\operatorname{Var}(W_t) = t`, :math:`\operatorname{Cov}(W_s,W_t) = \min(s,t)`.
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- **Quadratic variation:** :math:`[W]_T = T` (paths are non-differentiable but have finite :math:`p`-variation for :math:`p>2`).
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- **Self-similarity:** :math:`c^{-1/2}W_{ct} \overset{d}{=} W_t` (Hurst exponent :math:`H=\tfrac12`).
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**Example — Geometric BM:**
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:math:`S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)`
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is the Black–Scholes price model. Sample path sketch::
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S_t
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| .---.
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| .--./ \----.
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| / \---------.
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|/
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+-------------------------------> t
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0 T
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(log-normal marginals; continuous, nowhere-differentiable paths)
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### 2.2 Itô Calculus
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.. admonition:: Definition — Itô Integral
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For adapted :math:`f \in \mathcal{L}^2` (i.e. :math:`\mathbb{E}\!\int_0^T f_t^2\,dt < \infty`):
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.. math::
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\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}).
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The Itô integral is a **martingale** with zero mean and **Itô isometry**
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:math:`\mathbb{E}\bigl[(\int_0^T f_t\,dW_t)^2\bigr] = \mathbb{E}\int_0^T f_t^2\,dt`.
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.. admonition:: Theorem — Itô's Lemma
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For :math:`dX_t = \mu_t\,dt + \sigma_t\,dW_t` and :math:`F \in C^{1,2}([0,T]\times\mathbb{R})`:
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.. math::
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dF(t,X_t) = \partial_t F\,dt + \partial_x F\,dX_t + \tfrac{1}{2}\partial_{xx}F\,\sigma_t^2\,dt.
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The correction term :math:`\tfrac12\sigma^2\partial_{xx}F` (absent in ordinary calculus) arises
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from the non-zero quadratic variation :math:`d[W]_t = dt`.
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**Example:** Let :math:`X_t = \log S_t` with :math:`dS_t = \mu S_t\,dt + \sigma S_t\,dW_t`.
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Itô's Lemma gives :math:`dX_t = (\mu - \tfrac12\sigma^2)\,dt + \sigma\,dW_t`. ✔
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### 2.3 General Itô SDEs
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.. math::
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dX_t = b(t, X_t)\,dt + \boldsymbol{\sigma}(t, X_t)\,dW_t,\quad X_0 = x_0.
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**Existence & uniqueness (Picard–Lindelöf for SDEs):** If :math:`b, \boldsymbol{\sigma}` are
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globally Lipschitz with linear growth, there exists a unique strong solution with
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:math:`\mathbb{E}[\sup_{t\le T}\|X_t\|^2]<\infty`.
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.. list-table:: Common SDE Models
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:header-rows: 1
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:widths: 20 40 40
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* - Process
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- SDE
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- Stationary distribution
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* - Brownian motion
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- :math:`dX = \sigma\,dW`
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- —
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* - Geometric BM
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- :math:`dX = \mu X\,dt + \sigma X\,dW`
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- log-normal
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* - Ornstein–Uhlenbeck
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- :math:`dX = \kappa(\theta-X)\,dt + \sigma\,dW`
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- :math:`\mathcal{N}(\theta, \sigma^2/2\kappa)`
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* - CIR
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- :math:`dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW`
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- Gamma(:math:`2\kappa\theta/\sigma^2`, :math:`\sigma^2/2\kappa`)
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### 2.4 Ornstein-Uhlenbeck (Mean-Reversion)
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Used in Optimiz-rs's ``sparse_mean_reversion`` and ``ou_estimator`` modules:
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.. math::
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dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t.
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**Closed-form solution:**
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.. math::
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X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.
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**Half-life:** :math:`\tau_{1/2} = \ln 2/\kappa`. With :math:`\kappa=0.2`/day,
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half-life ≈ 3.5 days — typical for equity-pair spreads.
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**MLE log-likelihood** (discrete observations at spacing :math:`\Delta t`):
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.. math::
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\ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[\log(2\pi\hat\sigma_i^2)
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+ \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma_i^2}\right],
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where :math:`\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}` and
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:math:`\hat\sigma_i^2 = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})`.
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---
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## 3 · Jump Processes
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Many financial time series exhibit sudden large moves that Brownian motion cannot capture.
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### 3.1 Poisson Process
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.. admonition:: Definition — Poisson Process
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A counting process :math:`N = (N_t)_{t\ge 0}` is a *Poisson process with
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intensity* :math:`\lambda > 0` if:
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1. :math:`N_0 = 0`.
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2. Independent, stationary increments.
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3. :math:`\mathbb{P}(N_{t+h}-N_t=1) = \lambda h + o(h)` and :math:`\mathbb{P}(\Delta N > 1) = o(h)`.
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Equivalently, :math:`N_t \sim \text{Poisson}(\lambda t)` and inter-arrival times are
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:math:`\text{Exp}(\lambda)`. The *compensated* process :math:`\tilde N_t = N_t - \lambda t`
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is a martingale.
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### 3.2 Compound Poisson Jump-Diffusion (Merton 1976)
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.. math::
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\frac{dS_t}{S_{t^-}} = \mu\,dt + \sigma\,dW_t + d\Bigl(\sum_{k=1}^{N_t}(e^{J_k}-1)\Bigr),
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with :math:`N_t` Poisson(:math:`\lambda`) and :math:`J_k \sim \mathcal{N}(\mu_J, \sigma_J^2)`.
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**Merton option price** — a Poisson mixture of Black–Scholes prices:
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.. math::
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C_{\text{Merton}} = \sum_{n=0}^\infty \frac{e^{-\lambda' T}(\lambda' T)^n}{n!}
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\cdot C_{\text{BS}}\!\left(S_0, K, T, r_n, \sigma_n^2\right),
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where :math:`\lambda' = \lambda e^{\mu_J+\frac12\sigma_J^2}`,
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:math:`r_n = r - \lambda(e^{\mu_J+\frac12\sigma_J^2}-1) + n(\mu_J+\tfrac12\sigma_J^2)/T`,
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and :math:`\sigma_n^2 = \sigma^2 + n\sigma_J^2/T`.
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### 3.3 Lévy Processes and the Lévy–Khintchine Representation
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.. admonition:: Theorem — Lévy–Khintchine
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Every Lévy process (independent stationary increments) has characteristic function
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.. math::
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\mathbb{E}[e^{i\xi X_t}] = \exp\!\Bigl(t\Bigl[i b\xi - \tfrac{1}{2}\sigma^2\xi^2
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+ \int_{\mathbb{R}\setminus\{0\}} \bigl(e^{i\xi z}-1-i\xi z\mathbf{1}_{|z|\le1}\bigr)\nu(dz)\Bigr]\Bigr)
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where :math:`(b, \sigma^2, \nu)` is the *Lévy triplet* and :math:`\nu` the *Lévy measure*,
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satisfying :math:`\int(1\wedge z^2)\nu(dz)<\infty`.
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.. list-table:: Lévy Process Zoo
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:header-rows: 1
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:widths: 25 35 40
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* - Process
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- Lévy measure :math:`\nu`
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- Use case
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* - Brownian motion
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- :math:`\nu=0`
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- continuous diffusion
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* - Compound Poisson
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- finite measure
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- rare large jumps
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* - Variance Gamma
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- :math:`\nu(dz)\propto e^{-c|z|}/|z|`
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- equity returns
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* - CGMY
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||||
- :math:`e^{-G|z|}/|z|^{1+Y}` (neg), :math:`e^{-Mx}/x^{1+Y}` (pos)
|
||||
- heavy tails, :math:`Y\in(0,2)`
|
||||
* - :math:`\alpha`-stable
|
||||
- :math:`c|z|^{-1-\alpha}`
|
||||
- infinite-variance regimes
|
||||
|
||||
### 3.4 SDEs with Jumps — Generator and Itô Formula
|
||||
|
||||
.. math::
|
||||
|
||||
dX_t = b(X_{t^-})\,dt + \sigma(X_{t^-})\,dW_t
|
||||
+ \int_{\mathbb{R}} c(X_{t^-}, z)\,\tilde N(dt, dz),
|
||||
|
||||
where :math:`\tilde N(dt,dz) = N(dt,dz) - \nu(dz)\,dt` is the *compensated jump measure*.
|
||||
|
||||
**Itô formula for jump-diffusions:**
|
||||
|
||||
.. math::
|
||||
|
||||
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 *generator* is
|
||||
|
||||
.. math::
|
||||
|
||||
\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).
|
||||
|
||||
---
|
||||
|
||||
## 4 · Optimal Control (HJB, PMP, Jumps)
|
||||
|
||||
### 4.1 Stochastic HJB
|
||||
|
||||
For :math:`dX_t = b(X_t,u_t)\,dt + \sigma(X_t,u_t)\,dW_t`, minimising
|
||||
:math:`J = \mathbb{E}[\int_0^T \ell\,dt + g(X_T)]`, the value function
|
||||
:math:`V(t,x) = \inf_u J` satisfies:
|
||||
|
||||
.. math::
|
||||
|
||||
\boxed{
|
||||
-\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.
|
||||
}
|
||||
|
||||
Under smooth :math:`V`, the feedback law is
|
||||
:math:`u^\star(t,x) = \arg\min_u[\ell(x,u)+\nabla_x V^\top b(x,u)]`.
|
||||
|
||||
**LQR special case** (:math:`\ell = x^\top Q x + u^\top R u`, :math:`b=Ax+Bu`):
|
||||
:math:`V(t,x)=x^\top P(t)x + v(t)` with :math:`P` solving the *matrix Riccati ODE*:
|
||||
|
||||
.. math::
|
||||
|
||||
-\dot P = A^\top P + PA - PBR^{-1}B^\top P + Q,\quad P(T)=Q_T.
|
||||
|
||||
### 4.2 Pontryagin Maximum Principle
|
||||
|
||||
The PMP avoids the curse of dimensionality — it converts the HJB PDE into a
|
||||
two-point boundary-value problem in :math:`(X_t, p_t)`.
|
||||
|
||||
.. admonition:: Theorem (PMP)
|
||||
|
||||
Define the Hamiltonian :math:`\mathcal{H}(x,u,p) = \ell(x,u)+p^\top b(x,u)`.
|
||||
If :math:`(X^\star, u^\star)` is optimal, there exists a costate process :math:`p_t` with:
|
||||
|
||||
.. math::
|
||||
|
||||
\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 :math:`u_t^\star = \arg\min_u \mathcal{H}(X_t^\star, u, p_t)` holds a.e.
|
||||
|
||||
The costate pair :math:`(X_t^\star, p_t)` moves along Hamiltonian geodesics on
|
||||
:math:`T^\star\mathbb{R}^d` — a direct link to symplectic geometry (§10.4).
|
||||
|
||||
### 4.3 HJB with Jumps (HJBI)
|
||||
|
||||
Adding the jump term from §3.4, the HJB equation gains a non-local integral operator:
|
||||
|
||||
.. math::
|
||||
|
||||
-\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)-V(x)-\nabla V^\top c\bigr]\nu(dz)}_{\text{non-local jump term}}\Bigr].
|
||||
|
||||
Optimiz-rs's ``optimal_control`` module discretises the integral on a truncated support
|
||||
:math:`[-z_{\max}, z_{\max}]` using Gaussian quadrature.
|
||||
|
||||
### 4.4 Viscosity Solutions
|
||||
|
||||
When :math:`V` fails to be :math:`C^{1,2}` (degenerate :math:`\sigma`, state constraints),
|
||||
viscosity solutions (Crandall–Lions 1983) restore uniqueness:
|
||||
|
||||
.. admonition:: Definition — Viscosity Subsolution
|
||||
|
||||
A continuous :math:`V` is a viscosity *subsolution* if for every smooth :math:`\phi`
|
||||
touching :math:`V` from above at :math:`(t_0,x_0)`:
|
||||
:math:`-\partial_t\phi(t_0,x_0) \le \inf_u[\ldots]` evaluated at :math:`\phi`.
|
||||
|
||||
Optimiz-rs's backward DP converges to the viscosity solution under the CFL condition
|
||||
:math:`\Delta t \le C\,(\Delta x)^2`.
|
||||
|
||||
**Backward DP grid schema**::
|
||||
|
||||
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)
|
||||
|
||||
---
|
||||
|
||||
## 5 · Mean Field Games (1D Solver)
|
||||
|
||||
MFG couples a **backward HJB** (individual value) with a **forward Fokker–Planck** (population density):
|
||||
|
||||
.. math::
|
||||
|
||||
\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}
|
||||
|
||||
**Coupling:** :math:`H` depends on :math:`m` (mean-field interaction), creating a fixed-point problem.
|
||||
|
||||
**Fixed-point algorithm**::
|
||||
|
||||
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
|
||||
|
||||
**Convergence:** For monotone coupling (Lasry–Lions 2007), the system has a unique solution
|
||||
and the fixed-point iteration contracts.
|
||||
|
||||
**Practical tip:** Monitor both :math:`\|m^{k+1}-m^k\|_1` and :math:`\|u^{k+1}-u^k\|_\infty`;
|
||||
divergence of either signals non-monotone coupling or too large a time step.
|
||||
|
||||
---
|
||||
|
||||
## 6 · Kalman Filtering
|
||||
|
||||
### 6.1 Linear-Gaussian State Space
|
||||
|
||||
.. math::
|
||||
|
||||
\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).
|
||||
|
||||
Two-step recursion:
|
||||
|
||||
.. math::
|
||||
|
||||
\text{Predict:}\quad \hat{\mathbf{x}}^-_t = F\hat{\mathbf{x}}_{t-1},\quad P^-_t = FP_{t-1}F^\top+Q.
|
||||
|
||||
.. math::
|
||||
|
||||
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.
|
||||
|
||||
:math:`K_t` is the *Kalman gain* — it interpolates between full prior trust (:math:`K\to0`)
|
||||
and full observation trust (:math:`K\to H^{-1}`).
|
||||
|
||||
### 6.2 Information-Theoretic View
|
||||
|
||||
The Kalman filter computes the exact conditional mean
|
||||
:math:`\hat{\mathbf{x}}_t = \mathbb{E}[\mathbf{x}_t \mid \mathbf{y}_{1:t}]` in Gaussian models
|
||||
and minimises :math:`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)
|
||||
|
||||
For :math:`d\mathbf{X}_t = A\mathbf{X}_t\,dt + B\,d\mathbf{W}_t`,
|
||||
:math:`d\mathbf{Y}_t = C\mathbf{X}_t\,dt + d\mathbf{V}_t`, the error covariance satisfies
|
||||
the *Riccati ODE*:
|
||||
|
||||
.. math::
|
||||
|
||||
\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.
|
||||
|
||||
---
|
||||
|
||||
## 7 · MCMC (Metropolis–Hastings and Langevin)
|
||||
|
||||
### 7.1 Metropolis–Hastings
|
||||
|
||||
For target :math:`\pi(x) \propto e^{-U(x)}` and proposal :math:`q(x'\mid x)`:
|
||||
|
||||
.. math::
|
||||
|
||||
\alpha(x\to x') = \min\!\Bigl(1, \frac{\pi(x')q(x\mid x')}{\pi(x)q(x'\mid x)}\Bigr).
|
||||
|
||||
**Detailed balance** :math:`\pi(x)\alpha(x\to x') = \pi(x')\alpha(x'\to x)`
|
||||
ensures :math:`\pi` is the unique stationary distribution.
|
||||
|
||||
**Optimal scaling:** With Gaussian proposal :math:`q(x'|x)=\mathcal{N}(x,h^2 I_d)`,
|
||||
step :math:`h^\star \approx 2.38/\sqrt{d}` (Roberts–Gelman–Gilks 1997) targets ~23–45 % acceptance.
|
||||
|
||||
### 7.2 Langevin Dynamics (MALA)
|
||||
|
||||
Metropolis-Adjusted Langevin proposal:
|
||||
|
||||
.. math::
|
||||
|
||||
x' = x - \tfrac{h^2}{2}\nabla U(x) + h\,\xi, \quad \xi\sim\mathcal{N}(0,I_d),
|
||||
|
||||
a discretisation of the *overdamped Langevin SDE*:
|
||||
|
||||
.. math::
|
||||
|
||||
dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t,
|
||||
|
||||
whose stationary distribution is exactly :math:`\pi \propto e^{-U}` (Fokker–Planck analysis).
|
||||
|
||||
MALA converges in :math:`O(d^{1/3})` steps vs :math:`O(d)` for RW-MH — a key advantage
|
||||
for high-dimensional posteriors.
|
||||
|
||||
**Heuristic:** Tune proposal std so acceptance is ~25–45 %; see ``examples/notebooks/02_mcmc.ipynb`` for trace plots.
|
||||
|
||||
---
|
||||
|
||||
## 8 · Hidden Markov Models (HMM)
|
||||
|
||||
### 8.1 Model
|
||||
|
||||
Latent Markov chain :math:`Z_t \in \{1,\ldots,K\}` with transition matrix
|
||||
:math:`A_{ij}=\mathbb{P}(Z_t=j\mid Z_{t-1}=i)` generates observations
|
||||
:math:`Y_t \mid Z_t=k \sim B_k(y)`.
|
||||
|
||||
### 8.2 Baum–Welch (EM)
|
||||
|
||||
**E-step (forward–backward):**
|
||||
|
||||
.. math::
|
||||
|
||||
\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).
|
||||
|
||||
.. math::
|
||||
|
||||
\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}}.
|
||||
|
||||
**M-step:**
|
||||
|
||||
.. math::
|
||||
|
||||
\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
|
||||
iteration monotonically increases :math:`\mathcal{L}(\theta)` by Jensen's inequality.
|
||||
|
||||
**Viterbi (MAP path):** Replace sum-product with max-product:
|
||||
:math:`\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)`, runs in :math:`O(TK^2)`.
|
||||
|
||||
**Quality check:** Log-likelihood per EM iteration must be non-decreasing; a confusion matrix
|
||||
of Viterbi labels vs. ground truth validates regime recovery.
|
||||
|
||||
---
|
||||
|
||||
## 9 · Information Theory
|
||||
|
||||
### 9.1 Entropy and KL Divergence
|
||||
|
||||
.. admonition:: Definition — KL Divergence
|
||||
|
||||
For densities :math:`p, q`:
|
||||
|
||||
.. math::
|
||||
|
||||
D_{\mathrm{KL}}(p\,\|\,q) = \int p(x)\log\frac{p(x)}{q(x)}\,dx \;\ge\; 0,
|
||||
|
||||
with equality iff :math:`p=q` a.e. (Gibbs' inequality). Non-symmetric.
|
||||
|
||||
**Connection to model selection:** AIC :math:`= 2k - 2\ln\hat{\mathcal{L}}` and
|
||||
BIC :math:`= k\ln n - 2\ln\hat{\mathcal{L}}` bound :math:`D_{\mathrm{KL}}(p_{\text{true}}\,\|\,p_\theta)`.
|
||||
|
||||
### 9.2 Fisher Information
|
||||
|
||||
.. admonition:: Definition — Fisher Information Matrix
|
||||
|
||||
For parametric model :math:`p(x;\theta)`:
|
||||
|
||||
.. math::
|
||||
|
||||
\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].
|
||||
|
||||
**Cramér–Rao bound:** Any unbiased estimator :math:`\hat\theta` satisfies
|
||||
:math:`\operatorname{Cov}(\hat\theta) \succeq \mathcal{I}(\theta)^{-1}`.
|
||||
MLE achieves equality asymptotically.
|
||||
|
||||
**Example — Gaussian HMM emission** :math:`B_k = \mathcal{N}(\mu_k,\sigma_k^2)`:
|
||||
:math:`\mathcal{I}(\mu_k)=\sigma_k^{-2}`, :math:`\mathcal{I}(\sigma_k^2)=(2\sigma_k^4)^{-1}`.
|
||||
|
||||
### 9.3 Mutual Information and Feature Relevance
|
||||
|
||||
.. math::
|
||||
|
||||
I(X;Y) = D_{\mathrm{KL}}\bigl(p(X,Y)\,\|\,p(X)p(Y)\bigr) = H(X) - H(X\mid Y) \ge 0.
|
||||
|
||||
**mRMR criterion** (minimum redundancy, maximum relevance) for the sparse module:
|
||||
|
||||
.. math::
|
||||
|
||||
\max_{Y_i} \Bigl[I(Y_i;\text{target}) - \frac{1}{|S|}\sum_{Y_j\in S}I(Y_i;Y_j)\Bigr].
|
||||
|
||||
### 9.4 Natural Gradient (Preview)
|
||||
|
||||
Classical gradient descent ignores the geometry of parameter space. The *natural gradient*
|
||||
replaces :math:`\nabla_\theta\mathcal{L}` with :math:`\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}`,
|
||||
giving a reparametrisation-invariant update — see §10.2 for the full geometric development.
|
||||
|
||||
---
|
||||
|
||||
## 10 · Differential Geometry
|
||||
|
||||
### 10.1 Riemannian Manifolds
|
||||
|
||||
.. admonition:: Definition — Riemannian Manifold
|
||||
|
||||
A *Riemannian manifold* :math:`(M, g)` is a smooth manifold :math:`M` with a
|
||||
*metric tensor* :math:`g_p`: a symmetric, positive-definite bilinear form on each
|
||||
tangent space :math:`T_p M`.
|
||||
|
||||
**Geodesics** (locally shortest paths) satisfy:
|
||||
|
||||
.. math::
|
||||
|
||||
\ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0,
|
||||
|
||||
where :math:`\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
|
||||
|
||||
The statistical manifold :math:`\mathcal{M} = \{p(\cdot;\theta)\}` carries the
|
||||
**Fisher–Rao metric** :math:`g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}`.
|
||||
|
||||
**Natural gradient (Amari 1998):** Steepest descent on :math:`(\mathcal{M}, g)`:
|
||||
|
||||
.. math::
|
||||
|
||||
\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:**
|
||||
:math:`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.
|
||||
|
||||
**Dually flat structure:** Exponential families
|
||||
:math:`p(x;\theta)=h(x)\exp(\theta^\top T(x)-A(\theta))`
|
||||
are :math:`e`-flat in natural parameters and :math:`m`-flat in mean parameters
|
||||
:math:`\eta=\nabla A(\theta)`, with vanishing sectional curvature :math:`K=0` —
|
||||
explaining exact Newton/natural-gradient convergence on these models.
|
||||
|
||||
### 10.3 Lie Groups and Geometric Control
|
||||
|
||||
.. admonition:: Definition — Lie Group
|
||||
|
||||
A *Lie group* :math:`G` is a smooth manifold with a group structure where
|
||||
multiplication and inversion are smooth. The *Lie algebra* :math:`\mathfrak{g} = T_e G`
|
||||
linearises the group at the identity.
|
||||
|
||||
**Examples:**
|
||||
|
||||
- :math:`SO(d)` — rotation group; portfolio factor rotation and orthogonality constraints.
|
||||
- Heisenberg group — path-signature feature maps (used in ``lab_signature_methods``).
|
||||
|
||||
**Left-invariant control system on :math:`G`:**
|
||||
|
||||
.. math::
|
||||
|
||||
\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),
|
||||
providing structure-preserving optimal trajectories.
|
||||
|
||||
### 10.4 Symplectic Geometry and Hamiltonian Structure
|
||||
|
||||
The phase space :math:`(T^\star M, \omega)` carries the symplectic 2-form
|
||||
:math:`\omega = \sum_i dp_i \wedge dq_i`. Hamilton's equations preserve :math:`\omega`
|
||||
(*Liouville's theorem* — phase-space volume conserved).
|
||||
|
||||
**Connection to PMP:** The costate pair :math:`(X_t^\star, p_t)` solves Hamilton's equations,
|
||||
i.e., the PMP is a symplectic flow on :math:`T^\star\mathbb{R}^d`.
|
||||
|
||||
**Symplectic integrators** (Störmer–Verlet, Ruth–Forest) preserve :math:`\omega` discretely,
|
||||
keeping the Hamiltonian nearly constant over long horizons — critical for multi-year
|
||||
allocation back-tests in Optimiz-rs.
|
||||
|
||||
### 10.5 Sectional Curvature and Landscape Geometry
|
||||
|
||||
The sectional curvature :math:`K(\sigma)` governs how quickly nearby geodesics diverge::
|
||||
|
||||
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
|
||||
|
||||
For exponential families in natural/mean parameters :math:`K=0` — explaining exact
|
||||
Newton convergence without curvature correction.
|
||||
|
||||
---
|
||||
|
||||
## Quick Reference
|
||||
|
||||
.. list-table::
|
||||
:header-rows: 1
|
||||
:widths: 28 44 28
|
||||
|
||||
* - Concept
|
||||
- Key equation / object
|
||||
- Optimiz-rs module
|
||||
* - Brownian motion
|
||||
- :math:`W_t - W_s \sim \mathcal{N}(0,t-s)`
|
||||
- ``point_processes``
|
||||
* - Itô SDE
|
||||
- :math:`dX=b\,dt+\sigma\,dW`
|
||||
- ``ou_estimator``
|
||||
* - Poisson / Compound Poisson
|
||||
- :math:`N_t\sim\text{Poisson}(\lambda t)`
|
||||
- ``point_processes``
|
||||
* - Lévy process
|
||||
- triplet :math:`(b,\sigma^2,\nu)`
|
||||
- ``point_processes``
|
||||
* - HJB PDE
|
||||
- :math:`-\partial_t V = \inf_u[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}\sigma\sigma^\top\nabla^2 V]`
|
||||
- ``optimal_control``
|
||||
* - HJBI (jumps)
|
||||
- :math:`+\int[V(\cdot+c)-V-\nabla V^\top c]\nu\,dz`
|
||||
- ``optimal_control``
|
||||
* - PMP costate
|
||||
- :math:`\dot p = -\nabla_x\mathcal{H}`, :math:`u^\star=\arg\min_u\mathcal{H}`
|
||||
- ``optimal_control``
|
||||
* - MFG (HJB + KFP)
|
||||
- fixed-point :math:`u,m`
|
||||
- ``mean_field_games``
|
||||
* - Kalman filter
|
||||
- :math:`K_t = P^-H^\top(HP^-H^\top+R)^{-1}`
|
||||
- ``optimal_control``
|
||||
* - MALA
|
||||
- :math:`x'=x-\tfrac{h^2}{2}\nabla U+h\xi`
|
||||
- ``mcmc``
|
||||
* - HMM
|
||||
- Baum–Welch EM + Viterbi
|
||||
- ``hmm``
|
||||
* - Fisher information
|
||||
- :math:`\mathcal{I}_{ij}=\mathbb{E}[\partial_i\ell\,\partial_j\ell]`
|
||||
- ``hmm``, ``sparse``
|
||||
* - Natural gradient
|
||||
- :math:`\mathcal{I}^{-1}\nabla_\theta\mathcal{L}`
|
||||
- ``differential_evolution``
|
||||
* - Riemannian / Lie geometry
|
||||
- Christoffel symbols, Lie–Poisson equations
|
||||
- experimental
|
||||
* - DE (jDE)
|
||||
- mutation + crossover + selection
|
||||
- ``differential_evolution``
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. Øksendal, B. *Stochastic Differential Equations*, 6th ed. Springer, 2003.
|
||||
2. Cont, R. & Tankov, P. *Financial Modelling with Jump Processes*. CRC Press, 2004.
|
||||
3. Fleming, W.H. & Soner, H.M. *Controlled Markov Processes and Viscosity Solutions*. Springer, 2006.
|
||||
4. Lasry, J.-M. & Lions, P.-L. "Mean field games." *Jpn. J. Math.* **2** (2007) 229–260.
|
||||
5. Amari, S. *Information Geometry and Its Applications*. Springer, 2016.
|
||||
6. do Carmo, M.P. *Riemannian Geometry*. Birkhäuser, 1992.
|
||||
7. Holm, D.D., Marsden, J.E. & Ratiu, T.S. "The Euler–Poincaré equations." *Adv. Math.* **137** (1998).
|
||||
8. Price, K.V., Storn, R.M. & Lampinen, J.A. *Differential Evolution*. Springer, 2005.
|
||||
9. Roberts, G.O., Gelman, A. & Gilks, W.R. "Weak convergence of Metropolis algorithms." (1997).
|
||||
10. Merton, R.C. "Option pricing when underlying stock returns are discontinuous." *JFE* **3** (1976).
|
||||
11. Crandall, M.G. & Lions, P.-L. "Viscosity solutions of Hamilton–Jacobi equations." *Trans. AMS* (1983).
|
||||
|
||||
Reference in New Issue
Block a user