# 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 **definition block**, builds intuition through **examples**, and closes with a **notebook micro-check**. For complete walkthroughs see `examples/notebooks/`. --- ## 1 · Differential Evolution (DE) ### 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 $\{\mathbf{x}_{i,g}\} \subset \mathbb{R}^d$. **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. ### Operators | Step | Formula | Role | |------|---------|------| | 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 $U(0,1)s$. 3. $W_t - W_s \sim \mathcal{N}(0, t-s)$ for all $0\le s2$). - **Self-similarity:** $c^{-1/2}W_{ct} \overset{d}{=} W_t$ (Hurst exponent $H=\tfrac12$). **Example — Geometric BM:** $S_t = S_0 \exp\!\bigl((\mu-\tfrac12\sigma^2)t + \sigma W_t\bigr)$ is the Black–Scholes price model. Sample path sketch: ``` S_t | .---. | .--./ \----. | / \---------. |/ +-------------------------------> t 0 T (log-normal marginals; continuous, nowhere-differentiable paths) ``` ### 2.2 Itô Calculus ::::{admonition} Definition — Itô Integral :class: definition For adapted $f \in \mathcal{L}^2$ (i.e. $\mathbb{E}\!\int_0^T f_t^2\,dt < \infty$): $$\int_0^T f_t\,dW_t \;=\; L^2\text{-}\lim_{|\pi|\to 0} \sum_{k} f_{t_k}(W_{t_{k+1}}-W_{t_k}).$$ The Itô integral is a **martingale** with zero mean and **Itô isometry** $\mathbb{E}\bigl[(\int_0^T f_t\,dW_t)^2\bigr] = \mathbb{E}\int_0^T f_t^2\,dt$. :::: ::::{admonition} Theorem — Itô's Lemma :class: tip For $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and $F \in C^{1,2}([0,T]\times\mathbb{R})$: $$dF(t,X_t) = \partial_t F\,dt + \partial_x F\,dX_t + \tfrac{1}{2}\partial_{xx}F\,\sigma_t^2\,dt.$$ The correction term $\tfrac12\sigma^2\partial_{xx}F$ (absent in ordinary calculus) arises from the non-zero quadratic variation $d[W]_t = dt$. :::: **Example:** Let $X_t = \log S_t$ with $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$. Itô's Lemma gives $dX_t = (\mu - \tfrac12\sigma^2)\,dt + \sigma\,dW_t$. ✔ ### 2.3 General Itô SDEs $$dX_t = b(t, X_t)\,dt + \boldsymbol{\sigma}(t, X_t)\,dW_t,\quad X_0 = x_0.$$ **Existence & uniqueness (Picard–Lindelöf for SDEs):** If $b, \boldsymbol{\sigma}$ are globally Lipschitz with linear growth, there exists a unique strong solution with $\mathbb{E}[\sup_{t\le T}\|X_t\|^2]<\infty$. **Common SDE Models** | Process | SDE | Stationary distribution | |---------|-----|------------------------| | Brownian motion | $dX = \sigma\,dW$ | — | | Geometric BM | $dX = \mu X\,dt + \sigma X\,dW$ | log-normal | | Ornstein–Uhlenbeck | $dX = \kappa(\theta-X)\,dt + \sigma\,dW$ | $\mathcal{N}(\theta, \sigma^2/2\kappa)$ | | CIR | $dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW$ | Gamma$(2\kappa\theta/\sigma^2, \sigma^2/2\kappa)$ | ### 2.4 Ornstein-Uhlenbeck (Mean-Reversion) Used in Optimiz-rs's `sparse_mean_reversion` and `ou_estimator` modules: $$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t.$$ **Closed-form solution:** $$X_t = \theta + (X_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.$$ **Half-life:** $\tau_{1/2} = \ln 2/\kappa$. With $\kappa=0.2$/day, half-life ≈ 3.5 days — typical for equity-pair spreads. **MLE log-likelihood** (discrete observations at spacing $\Delta t$): $$\ell(\kappa,\theta,\sigma) = -\frac{1}{2}\sum_{i=1}^{n}\left[\log(2\pi\hat\sigma_i^2) + \frac{(X_{t_i} - \hat\mu_i)^2}{\hat\sigma_i^2}\right],$$ where $\hat\mu_i = \theta + (X_{t_{i-1}}-\theta)e^{-\kappa\Delta t}$ and $\hat\sigma_i^2 = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa\Delta t})$. --- ## 3 · Jump Processes Many financial time series exhibit sudden large moves that Brownian motion cannot capture. ### 3.1 Poisson Process ::::{admonition} Definition — Poisson Process :class: definition 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 $\mathbb{P}(\Delta N > 1) = o(h)$. :::: Equivalently, $N_t \sim \text{Poisson}(\lambda t)$ and inter-arrival times are $\text{Exp}(\lambda)$. The *compensated* process $\tilde N_t = N_t - \lambda t$ is a martingale. ### 3.2 Compound Poisson Jump-Diffusion (Merton 1976) $$\frac{dS_t}{S_{t^-}} = \mu\,dt + \sigma\,dW_t + d\Bigl(\sum_{k=1}^{N_t}(e^{J_k}-1)\Bigr),$$ with $N_t$ Poisson($\lambda$) and $J_k \sim \mathcal{N}(\mu_J, \sigma_J^2)$. **Merton option price** — a Poisson mixture of Black–Scholes prices: $$C_{\text{Merton}} = \sum_{n=0}^\infty \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} \cdot C_{\text{BS}}\!\left(S_0, K, T, r_n, \sigma_n^2\right),$$ 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 $\sigma_n^2 = \sigma^2 + n\sigma_J^2/T$. ### 3.3 Lévy Processes and the Lévy–Khintchine Representation ::::{admonition} Theorem — Lévy–Khintchine :class: tip Every Lévy process (independent stationary increments) has characteristic function $$\mathbb{E}[e^{i\xi X_t}] = \exp\!\Bigl(t\Bigl[i b\xi - \tfrac{1}{2}\sigma^2\xi^2 + \int_{\mathbb{R}\setminus\{0\}} \bigl(e^{i\xi z}-1-i\xi z\mathbf{1}_{|z|\le1}\bigr)\nu(dz)\Bigr]\Bigr)$$ where $(b, \sigma^2, \nu)$ is the *Lévy triplet* and $\nu$ the *Lévy measure*, satisfying $\int(1\wedge z^2)\nu(dz)<\infty$. :::: **Lévy Process Zoo** | Process | Lévy measure $\nu$ | Use case | |---------|-------------------|----------| | Brownian motion | $\nu=0$ | continuous diffusion | | Compound Poisson | finite measure | rare large jumps | | Variance Gamma | $\nu(dz)\propto e^{-c\|z\|}/\|z\|$ | equity returns | | CGMY | $e^{-G\|z\|}/\|z\|^{1+Y}$ (neg), $e^{-Mx}/x^{1+Y}$ (pos) | heavy tails, $Y\in(0,2)$ | | $\alpha$-stable | $c\|z\|^{-1-\alpha}$ | infinite-variance regimes | ### 3.4 SDEs with Jumps — Generator and Itô Formula $$dX_t = b(X_{t^-})\,dt + \sigma(X_{t^-})\,dW_t + \int_{\mathbb{R}} c(X_{t^-}, z)\,\tilde N(dt, dz),$$ where $\tilde N(dt,dz) = N(dt,dz) - \nu(dz)\,dt$ is the *compensated jump measure*. **Itô formula for jump-diffusions:** $$dF(X_t) = \mathcal{L}F\,dt + \partial_x F\,\sigma\,dW_t + \int\bigl[F(X_{t^-}+c)-F(X_{t^-})\bigr]\tilde N(dt,dz),$$ where the *generator* is $$\mathcal{L}F = b\,\partial_x F + \tfrac12\sigma^2\partial_{xx}F + \int\bigl[F(x+c)-F(x)-c\,\partial_x F\bigr]\nu(dz).$$ --- ## 4 · Optimal Control (HJB, PMP, Jumps) **Big picture.** Optimal control asks: *given a stochastic system we can steer with a control $u_t$, what policy minimises expected cost?* Three complementary tools answer this: | Tool | Solves | Scales to | Intuition | |------|--------|-----------|-----------| | 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 | --- ### 4.1 Stochastic HJB **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 **value function** $V(t,x) = \inf_u J(t,x;u)$ satisfies: $$-\partial_t V = \inf_{u\in\mathcal{U}}\Bigl[\ell(x,u) + \nabla_x V^{\!\top} b(x,u) + \tfrac12\operatorname{Tr}\bigl(\sigma\sigma^{\!\top}(x,u)\,\nabla_x^2 V\bigr)\Bigr], \quad V(T,\cdot)=g.$$ **Intuition.** The three terms inside the infimum are: - $\ell(x,u)$ — instantaneous running cost (pay now), - $\nabla_x V^\top b$ — drift of the value (first-order) - $\tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2 V)$ — curvature correction due to noise (the stochastic analogue of a second-order Taylor term). Under smooth $V$, the **feedback law** is $u^\star(t,x) = \arg\min_u[\ell(x,u)+\nabla_x V^\top b(x,u)].$ --- ::::{admonition} Example — Optimal Portfolio Allocation :class: note 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: $$u^\star = \frac{\mu-r}{\sigma^2} \quad (\text{fraction in risky asset}).$$ This is the classic Merton (1969) result: invest a fixed fraction proportional to the Sharpe ratio and inversely to variance — independent of wealth and time. :::: --- **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 *matrix Riccati ODE*: $$-\dot P = A^\top P + PA - PBR^{-1}B^\top P + Q,\quad P(T)=Q_T.$$ The optimal control is **linear feedback**: $u^\star_t = -R^{-1}B^\top P(t)X_t$. --- ::::{admonition} Example — Optimal Inventory (Almgren–Chriss liquidation) :class: note A trader must liquidate $X_0$ shares by time $T$. Inventory $X_t$, trading rate $u_t<0$: $$dX_t = u_t\,dt, \quad \ell(x,u) = \underbrace{\alpha x^2}_{\text{risk}} + \underbrace{\beta u^2}_{\text{impact}}.$$ This is a **deterministic LQR** ($\sigma=0$) with $A=0$, $B=1$, $Q=\alpha$, $R=\beta$. 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. :::: --- ### 4.2 Pontryagin Maximum Principle The PMP avoids the curse of dimensionality — it converts the HJB PDE into a **two-point boundary-value ODE** in $(X_t, p_t)$, making it feasible in high dimensions where a PDE grid is intractable. ::::{admonition} Theorem (PMP) :class: tip Define the **Hamiltonian** $\mathcal{H}(x,u,p) = \ell(x,u)+p^\top b(x,u)$. If $(X^\star, u^\star)$ is optimal, there exists a **costate** (adjoint) process $p_t$ with: $$\dot p_t = -\nabla_x \mathcal{H}(X_t^\star, u_t^\star, p_t),\quad p_T = \nabla_x g(X_T^\star),$$ and the optimality condition $u_t^\star = \arg\min_u \mathcal{H}(X_t^\star, u, p_t)$ holds a.e. :::: **Costate intuition.** $p_t$ is the *shadow price* of state $X_t$: $$p_t = \nabla_x V(t, X_t^\star) = \frac{\partial (\text{optimal cost-to-go})}{\partial x}.$$ Increasing the current state by $dx$ changes future cost by $p_t^\top dx$. This is exactly the adjoint/backpropagation equation of deep learning — PMP is the continuous-time version of gradient backpropagation through a dynamical system. **Algorithm (shooting method):** ``` 1. Guess costate p_0 2. Integrate forward: dX = b(X, u*(X,p)) dt (state ODE) 3. Integrate backward: dp = -∇_x H(X, u*, p) dt (costate ODE) 4. Check boundary condition: p_T = ∇g(X_T) 5. If not satisfied -> update p_0 (Newton / gradient) -> go to 2 ``` --- ::::{admonition} Example — PMP for the Merton Problem :class: note With $\ell = 0$, $g(x) = -\ln x$, $b = (r+u(\mu-r))x$, $\sigma^\top\sigma = u^2\sigma^2 x^2$, the Hamiltonian is $\mathcal{H}(x,u,p) = p(r+u(\mu-r))x$. **Costate ODE:** $\dot p_t = -\partial_x \mathcal{H} = -p_t(r+u^\star(\mu-r))$, with terminal $p_T = -1/X_T^\star$. **Optimality condition** $\partial_u\mathcal{H}=0$ gives $p_t(\mu-r)x + \partial_u(\tfrac12\sigma^2 u^2 x^2 \partial_{xx}V)=0$, recovering $u^\star = (\mu-r)/\sigma^2$ as before. 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. :::: --- 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). --- ### 4.3 HJB with Jumps (HJBI) When the state can jump (§3.4), the HJB equation gains a **non-local integral operator**: $$-\partial_t V = \inf_{u}\Bigl[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2 V) + \underbrace{\int\bigl[V(x+c(x,u,z))-V(x)-\nabla V^\top c(x,u,z)\bigr]\nu(dz)}_{\text{expected value change from jumps}}\Bigr].$$ **Intuition for the integral term.** A jump of size $c$ moves the state from $x$ to $x+c$, changing the value function by $V(x+c)-V(x)$. The compensator $\nabla V^\top c$ subtracts the linear part already counted in the drift, following Itô's formula for jump processes (§3.4). The `optimal_control` module discretises the integral on a truncated support $[-z_{\max}, z_{\max}]$ using Gaussian quadrature. --- ::::{admonition} Example — Optimal Execution with Jump Risk :class: note Extend the inventory model with Poisson order-flow shocks: $$dX_t = u_t\,dt + \Delta J_t,\quad \Delta J_t \sim \text{Compound Poisson}(\lambda, \mathcal{N}(0,\sigma_J^2)).$$ The HJBI becomes: $$-\partial_t V = \inf_u\Bigl[\alpha x^2 + \beta u^2 + \partial_x V\,u + \lambda\,\mathbb{E}_z[V(x+z)-V(x)-z\,\partial_x V]\Bigr].$$ With Gaussian jumps, the expectation computes as $\lambda(\tfrac12\sigma_J^2\,\partial_{xx}V)$, so the HJBI reduces to the same LQR Riccati ODE but with **effective diffusion** $\sigma_{\text{eff}}^2 = \lambda\sigma_J^2$. Key insight: order-flow risk acts like additional Brownian volatility, accelerating the optimal sell schedule. :::: --- ### 4.4 Viscosity Solutions When $V$ fails to be $C^{1,2}$ — which happens with degenerate diffusion ($\sigma \approx 0$), state/control constraints, or non-smooth terminal conditions — classical solutions may not exist. **Viscosity solutions** (Crandall–Lions 1983) provide a rigorous weak notion that restores existence and uniqueness. **Why they matter:** In practice, HJB is solved on a grid and $V$ is only piecewise smooth. Viscosity theory guarantees the numerical scheme converges to the true solution. ::::{admonition} Definition — Viscosity Subsolution :class: definition A continuous $V$ is a viscosity *subsolution* if for every smooth $\phi$ touching $V$ **from above** at $(t_0,x_0)$ (i.e., $V - \phi$ has a local maximum there): $$-\partial_t\phi(t_0,x_0) \le \inf_u\Bigl[\ell(x_0,u) + \nabla_x\phi^\top b + \tfrac12\operatorname{Tr}(\sigma\sigma^\top\nabla^2\phi)\Bigr].$$ A *supersolution* reverses the inequality with a smooth test touching from *below*. The unique viscosity **solution** is simultaneously both. :::: **Practical interpretation.** Classical calculus says "$V$ satisfies the PDE pointwise." Viscosity theory says "$V$ satisfies the PDE in an averaged sense via test functions — even at kinks." The condition prevents $V$ from being arbitrarily steep or flat at non-smooth points. Optimiz-rs's backward DP converges to the viscosity solution under the CFL condition $\Delta t \le C\,(\Delta x)^2$. --- ::::{admonition} Example — American Option as a Viscosity Problem :class: note An American put has early-exercise payoff $g(x) = (K-x)^+$. The value function satisfies the **variational inequality** (a two-region HJB): $$\min\Bigl(-\partial_t V - \mathcal{L}_{\text{BS}}V,\; V - (K-x)^+\Bigr) = 0,$$ where $\mathcal{L}_{\text{BS}}V = rx\partial_x V + \tfrac12\sigma^2 x^2\partial_{xx}V - rV$. - **Continuation region** ($V > (K-x)^+$): the Black–Scholes PDE holds. - **Exercise region** ($V = (K-x)^+$): the option is exercised immediately. At the free boundary the gradient $\partial_x V$ is continuous (*smooth-pasting*) but $\partial_{xx}V$ is not — so $V$ is only $C^1$, not $C^2$. Viscosity theory handles this kink rigorously. :::: **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): $$\begin{aligned} \text{HJB (backward): } & -\partial_t u - \nu\partial_{xx}u + H(x,\partial_x u, m) = 0, & u(T,x)&=g(x),\\ \text{Fokker–Planck (forward): } & \partial_t m - \nu\partial_{xx}m - \partial_x(m\,\partial_p H) = 0, & m(0,x)&=m_0(x). \end{aligned}$$ **Coupling:** $H$ depends on $m$ (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 $\|m^{k+1}-m^k\|_1$ and $\|u^{k+1}-u^k\|_\infty$; divergence of either signals non-monotone coupling or too large a time step. --- ## 6 · Kalman Filtering ### 6.1 Linear-Gaussian State Space $$\mathbf{x}_t = F\mathbf{x}_{t-1} + \mathbf{w}_t,\; \mathbf{w}_t\sim\mathcal{N}(0,Q); \qquad \mathbf{y}_t = H\mathbf{x}_t + \mathbf{v}_t,\; \mathbf{v}_t\sim\mathcal{N}(0,R).$$ **Predict:** $$\hat{\mathbf{x}}^-_t = F\hat{\mathbf{x}}_{t-1},\quad P^-_t = FP_{t-1}F^\top+Q.$$ **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 *Kalman gain* — it interpolates between full prior trust ($K\to0$) and full observation trust ($K\to H^{-1}$). ### 6.2 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. ### 6.3 Continuous-Time Limit (Kalman–Bucy) For $d\mathbf{X}_t = A\mathbf{X}_t\,dt + B\,d\mathbf{W}_t$, $d\mathbf{Y}_t = C\mathbf{X}_t\,dt + d\mathbf{V}_t$, the error covariance satisfies the *Riccati ODE*: $$\dot P = AP + PA^\top + BQB^\top - PC^\top R^{-1}CP,\qquad P(0)=P_0,$$ which converges to the algebraic Riccati solution at steady state. --- ## 7 · MCMC (Metropolis–Hastings and Langevin) ### 7.1 Metropolis–Hastings For target $\pi(x) \propto e^{-U(x)}$ and proposal $q(x'\mid x)$: $$\alpha(x\to x') = \min\!\Bigl(1, \frac{\pi(x')q(x\mid x')}{\pi(x)q(x'\mid x)}\Bigr).$$ **Detailed balance** $\pi(x)\alpha(x\to x') = \pi(x')\alpha(x'\to x)$ ensures $\pi$ is the unique stationary distribution. **Optimal scaling:** With Gaussian proposal $q(x'|x)=\mathcal{N}(x,h^2 I_d)$, step $h^\star \approx 2.38/\sqrt{d}$ (Roberts–Gelman–Gilks 1997) targets ~23–45 % acceptance. ### 7.2 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 *overdamped Langevin SDE*: $$dX_t = -\nabla U(X_t)\,dt + \sqrt{2}\,dW_t,$$ whose stationary distribution is exactly $\pi \propto e^{-U}$ (Fokker–Planck analysis). MALA converges in $O(d^{1/3})$ steps vs $O(d)$ 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 $Z_t \in \{1,\ldots,K\}$ with transition matrix $A_{ij}=\mathbb{P}(Z_t=j\mid Z_{t-1}=i)$ generates observations $Y_t \mid Z_t=k \sim B_k(y)$. ### 8.2 Baum–Welch (EM) **E-step (forward–backward):** $$\alpha_t(k) = B_k(y_t)\sum_j \alpha_{t-1}(j)A_{jk}, \qquad \beta_t(k) = \sum_j A_{kj}B_j(y_{t+1})\beta_{t+1}(j).$$ $$\gamma_t(k) = \frac{\alpha_t(k)\beta_t(k)}{\sum_j \alpha_t(j)\beta_t(j)}, \qquad \xi_t(j,k) = \frac{\alpha_t(j)A_{jk}B_k(y_{t+1})\beta_{t+1}(k)}{\mathcal{L}}.$$ **M-step:** $$\hat A_{jk} = \frac{\sum_t \xi_t(j,k)}{\sum_t\gamma_t(j)}, \qquad \hat\mu_k = \frac{\sum_t \gamma_t(k)\,y_t}{\sum_t \gamma_t(k)}.$$ **Information-theoretic view:** Baum–Welch is EM on the complete-data log-likelihood; each iteration monotonically increases $\mathcal{L}(\theta)$ by Jensen's inequality. **Viterbi (MAP path):** Replace sum-product with max-product: $\delta_t(k) = \max_j \delta_{t-1}(j)A_{jk} \cdot B_k(y_t)$, runs in $O(TK^2)$. **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 :class: definition 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. :::: **Connection to model selection:** AIC $= 2k - 2\ln\hat{\mathcal{L}}$ and BIC $= k\ln n - 2\ln\hat{\mathcal{L}}$ bound $D_{\mathrm{KL}}(p_{\text{true}}\,\|\,p_\theta)$. ### 9.2 Fisher Information ::::{admonition} Definition — Fisher Information Matrix :class: definition For parametric model $p(x;\theta)$: $$\mathcal{I}(\theta)_{ij} = \mathbb{E}_{x\sim p}\!\left[\partial_{\theta_i}\log p\;\partial_{\theta_j}\log p\right] = -\mathbb{E}\!\left[\partial^2_{\theta_i\theta_j}\log p\right].$$ :::: **Cramér–Rao bound:** Any unbiased estimator $\hat\theta$ satisfies $\operatorname{Cov}(\hat\theta) \succeq \mathcal{I}(\theta)^{-1}$. MLE achieves equality asymptotically. **Example — Gaussian HMM emission** $B_k = \mathcal{N}(\mu_k,\sigma_k^2)$: $\mathcal{I}(\mu_k)=\sigma_k^{-2}$, $\mathcal{I}(\sigma_k^2)=(2\sigma_k^4)^{-1}$. ### 9.3 Mutual Information and Feature Relevance $$I(X;Y) = D_{\mathrm{KL}}\bigl(p(X,Y)\,\|\,p(X)p(Y)\bigr) = H(X) - H(X\mid Y) \ge 0.$$ **mRMR criterion** (minimum redundancy, maximum relevance) for the sparse module: $$\max_{Y_i} \Bigl[I(Y_i;\text{target}) - \frac{1}{|S|}\sum_{Y_j\in S}I(Y_i;Y_j)\Bigr].$$ ### 9.4 Natural Gradient (Preview) Classical gradient descent ignores the geometry of parameter space. The *natural gradient* replaces $\nabla_\theta\mathcal{L}$ with $\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}$, giving a reparametrisation-invariant update — see §10.2 for the full geometric development. --- ## 10 · Differential Geometry ### 10.1 Riemannian Manifolds ::::{admonition} Definition — Riemannian Manifold :class: definition 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 $T_p M$. :::: **Geodesics** (locally shortest paths) satisfy: $$\ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}\,\dot\gamma^i\dot\gamma^j = 0,$$ where $\Gamma^k_{ij} = \tfrac12 g^{kl}(\partial_i g_{jl}+\partial_j g_{il}-\partial_l g_{ij})$ are the *Christoffel symbols* encoding intrinsic curvature. ### 10.2 Information Geometry and Fisher–Rao Metric The statistical manifold $\mathcal{M} = \{p(\cdot;\theta)\}$ carries the **Fisher–Rao metric** $g_{ij}(\theta) = \mathcal{I}(\theta)_{ij}$. **Natural gradient (Amari 1998):** Steepest descent on $(\mathcal{M}, g)$: $$\theta \leftarrow \theta - \eta\,\mathcal{I}(\theta)^{-1}\nabla_\theta\mathcal{L}.$$ This is *invariant to reparametrisation* and achieves quadratic convergence on convex objectives — equivalent to Fisher scoring. **KL geometry:** $D_{\mathrm{KL}}(p_\theta\,\|\,p_{\theta+d\theta}) = \tfrac12\,d\theta^\top\mathcal{I}(\theta)\,d\theta + O(\|d\theta\|^3)$, confirming Fisher–Rao as the intrinsic KL metric. **Dually flat structure:** Exponential families $p(x;\theta)=h(x)\exp(\theta^\top T(x)-A(\theta))$ are $e$-flat in natural parameters and $m$-flat in mean parameters $\eta=\nabla A(\theta)$, with vanishing sectional curvature $K=0$ — explaining exact Newton/natural-gradient convergence on these models. ### 10.3 Lie Groups and Geometric Control ::::{admonition} Definition — Lie Group :class: definition 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. :::: **Examples:** - $SO(d)$ — rotation group; portfolio factor rotation and orthogonality constraints. - Heisenberg group — path-signature feature maps (used in `lab_signature_methods`). **Left-invariant control system on $G$:** $$\dot g(t) = g(t)\,\xi(t), \quad g\in G,\; \xi(t)\in\mathfrak{g}.$$ PMP on Lie groups yields the *Lie–Poisson (Euler–Poincaré) equations* (Holm–Marsden–Ratiu), providing structure-preserving optimal trajectories. ### 10.4 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 $\omega$ (*Liouville's theorem* — phase-space volume conserved). **Connection to PMP:** The costate pair $(X_t^\star, p_t)$ solves Hamilton's equations, i.e., the PMP is a symplectic flow on $T^\star\mathbb{R}^d$. **Symplectic integrators** (Störmer–Verlet, Ruth–Forest) preserve $\omega$ discretely, 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 $K(\sigma)$ 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 $K=0$ — explaining exact Newton convergence without curvature correction. --- ## Quick Reference | Concept | Key equation / object | Optimiz-rs module | |---------|----------------------|-------------------| | Brownian motion | $W_t - W_s \sim \mathcal{N}(0,t-s)$ | `point_processes` | | Itô SDE | $dX=b\,dt+\sigma\,dW$ | `ou_estimator` | | Poisson / Compound Poisson | $N_t\sim\text{Poisson}(\lambda t)$ | `point_processes` | | Lévy process | triplet $(b,\sigma^2,\nu)$ | `point_processes` | | HJB PDE | $-\partial_t V = \inf_u[\ell + \nabla V^\top b + \tfrac12\operatorname{Tr}\sigma\sigma^\top\nabla^2 V]$ | `optimal_control` | | HJBI (jumps) | $+\int[V(\cdot+c)-V-\nabla V^\top c]\nu\,dz$ | `optimal_control` | | PMP costate | $\dot p = -\nabla_x\mathcal{H}$, $u^\star=\arg\min_u\mathcal{H}$ | `optimal_control` | | MFG (HJB + KFP) | fixed-point $u,m$ | `mean_field_games` | | Kalman filter | $K_t = P^-H^\top(HP^-H^\top+R)^{-1}$ | `optimal_control` | | MALA | $x'=x-\tfrac{h^2}{2}\nabla U+h\xi$ | `mcmc` | | HMM | Baum–Welch EM + Viterbi | `hmm` | | Fisher information | $\mathcal{I}_{ij}=\mathbb{E}[\partial_i\ell\,\partial_j\ell]$ | `hmm`, `sparse` | | Natural gradient | $\mathcal{I}^{-1}\nabla_\theta\mathcal{L}$ | `differential_evolution` | | Riemannian / Lie geometry | Christoffel symbols, Lie–Poisson equations | experimental | | 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).