optimiz-rs/docs/source/theory/mathematical_foundations.md

# 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 $U(0,1)<CR$ or $j=j_\text{rand}$ | mix dimensions |
| Greedy selection | $\mathbf{x}_{i,g+1} = \mathbf{u}_{i,g}$ iff $f(\mathbf{u})\le f(\mathbf{x})$ | exploit |

**Convergence (informal):** Under bounded population diversity and Lipschitz $f$, the
best-so-far value converges a.s. to a stationary point as $N,g\to\infty$ (Price et al. 2005).

### Self-Adaptive jDE (Optimiz-rs default)

Parameters $F,CR$ are per-individual and reset stochastically each generation:

$$
F_i^{g+1} = \begin{cases} F_{\min} + r_1 F_{\max} & r_2 < \tau_1,\\ F_i^g & \text{otherwise,}\end{cases}
\qquad
CR_i^{g+1} = \begin{cases} U(0,1) & r_3 < \tau_2,\\ CR_i^g & \text{otherwise.}\end{cases}
$$

$\tau_1=\tau_2=0.1$ by default.  On rugged landscapes this produces bimodal $F$
histograms concentrated near 0.8 — a sign the landscape is highly multimodal.

**Notebook check** (`05_performance_benchmarks.ipynb`): Plot $F_i, CR_i$ histograms
every 50 generations; expect values clustering in $[0.5,0.9]$ on hard problems.

---

## 2 · Stochastic Processes

These form the probabilistic backbone of all continuous-time models in Optimiz-rs.

### 2.1 Brownian Motion

::::{admonition} Definition — Wiener Process
:class: definition

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 $0\le s<t$.
4. Paths $t\mapsto W_t(\omega)$ are **continuous** a.s.
::::

**Key properties:**

- $\mathbb{E}[W_t] = 0$, $\operatorname{Var}(W_t) = t$, $\operatorname{Cov}(W_s,W_t) = \min(s,t)$.
- **Quadratic variation:** $[W]_T = T$ (paths are non-differentiable but have finite $p$-variation for $p>2$).
- **Self-similarity:** $c^{-1/2}W_{ct} \overset{d}{=} W_t$ (Hurst exponent $H=\tfrac12$).

**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).