docs(v2.0.0-alpha.6): convert all inline $...$ to :math: role in v2 RST pages

RST does not parse dollar-math (the dollarmath MyST extension applies
only to .md files), so every inline LaTeX expression was rendered as
raw text on Read the Docs — with backslashes silently stripped by the
RST escape mechanism (e.g. \bar s shown as 'bar s', \mathbb{E} shown
as 'mathbb{E}'). The eight v2.0 algorithm pages now use the proper
:math: role for inline math (224 expressions converted), so MathJax
renders every symbol correctly.

Affected pages: bsde, pde, stochastic_control, quadratic_impact_control,
mckean_vlasov, agent_based, robust_drift, generative_calibration_hooks.

docs/source/algorithms/agent_based.rst

@@ -10,53 +10,53 @@ update rule
    \qquad \bar s^k \;=\; \frac1N \sum_{j=1}^N s^k_j,
    \qquad \xi^k_i \sim \mathcal{N}(0, \sigma^2),
 
-with $\alpha \in (0, 1]$ the *averaging weight* and $\sigma$ the noise scale.  This is the
+with :math:`\alpha \in (0, 1]` the *averaging weight* and :math:`\sigma` the noise scale.  This is the
 DeGroot–Friedkin–Johnsen baseline of opinion dynamics, and the *complete-graph* limit of the
 Hegselmann–Krause and Vicsek flocking models.
 
 Mathematical background
 -----------------------
 
-**Mean conservation.**  Averaging the update over $i$ gives
+**Mean conservation.**  Averaging the update over :math:`i` gives
-$\bar s^{k+1} = \bar s^k + \bar\xi^k$ with $\mathbb{E}[\bar\xi^k] = 0$, so the empirical mean
+:math:`\bar s^{k+1} = \bar s^k + \bar\xi^k` with :math:`\mathbb{E}[\bar\xi^k] = 0`, so the empirical mean
 is a *martingale* and is exactly preserved in expectation:
 
 .. math::
 
    \mathbb{E}[\bar s^k] \;=\; \bar s^0 \quad \text{for all } k \ge 0.
 
-In the noiseless case $\sigma = 0$ the mean is preserved *path-by-path*.
+In the noiseless case :math:`\sigma = 0` the mean is preserved *path-by-path*.
 
-**Geometric contraction of the spread.**  Define the deviation $d^k_i := s^k_i - \bar s^k$.
+**Geometric contraction of the spread.**  Define the deviation :math:`d^k_i := s^k_i - \bar s^k`.
 The update implies
 
 .. math::
 
    d^{k+1}_i \;=\; (1 - \alpha)\, d^k_i \;+\; \bigl(\xi^k_i - \bar\xi^k\bigr) ,
 
-so in the absence of noise $\| d^k \|_\infty \le (1 - \alpha)^k \| d^0 \|_\infty$ — the spread
+so in the absence of noise :math:`\| d^k \|_\infty \le (1 - \alpha)^k \| d^0 \|_\infty` — the spread
-*contracts geometrically* with rate $1 - \alpha$.  The companion notebook plots
+*contracts geometrically* with rate :math:`1 - \alpha`.  The companion notebook plots
-$\max_i s^k_i - \min_i s^k_i$ on a log scale across $\alpha \in \{0.05, \dots, 1\}$ and
+:math:`\max_i s^k_i - \min_i s^k_i` on a log scale across :math:`\alpha \in \{0.05, \dots, 1\}` and
 recovers exactly this slope.
 
 **Stationary variance with noise.**  Treating the deviation as an AR(1) process with input
-variance $\sigma^2 (1 - 1/N)$, the steady-state variance of any single agent's deviation is
+variance :math:`\sigma^2 (1 - 1/N)`, the steady-state variance of any single agent's deviation is
 
 .. math::
 
    \mathrm{Var}_\infty(d_i) \;=\; \frac{\sigma^2 (1 - 1/N)}{1 - (1 - \alpha)^2}
      \;\xrightarrow[\alpha \to 0]{}\; \frac{\sigma^2}{2\alpha}\,(1 - 1/N).
 
-**Continuous-time limit (linear Vlasov).**  Sending $\alpha = \theta\, \Delta t$,
+**Continuous-time limit (linear Vlasov).**  Sending :math:`\alpha = \theta\, \Delta t`,
-$\xi^k_i = \sigma \sqrt{\Delta t}\, W^i_k$ and $\Delta t \to 0$ recovers the McKean–Vlasov SDE
+:math:`\xi^k_i = \sigma \sqrt{\Delta t}\, W^i_k` and :math:`\Delta t \to 0` recovers the McKean–Vlasov SDE
-$dX^i_t = \theta(\bar X_t - X^i_t)\, dt + \sigma\, dW^i_t$ of :doc:`mckean_vlasov` — the
+:math:`dX^i_t = \theta(\bar X_t - X^i_t)\, dt + \sigma\, dW^i_t` of :doc:`mckean_vlasov` — the
 discrete consensus update is the prototype of mean-field interaction.
 
 **Spectral interpretation.**  On a general weighted graph the update reads
-$s^{k+1} = (I - \alpha L)\, s^k + \xi^k$, where $L$ is the normalised Laplacian.  The
+:math:`s^{k+1} = (I - \alpha L)\, s^k + \xi^k`, where :math:`L` is the normalised Laplacian.  The
-complete-graph case shipped here has $L = I - \tfrac1N \mathbf{1}\mathbf{1}^\top$ with
+complete-graph case shipped here has :math:`L = I - \tfrac1N \mathbf{1}\mathbf{1}^\top` with
-eigenvalue $1$ on the orthogonal complement of $\mathbf{1}$, hence the contraction rate
+eigenvalue :math:`1` on the orthogonal complement of :math:`\mathbf{1}`, hence the contraction rate
-$1 - \alpha$ above.  Replacing $\mathbf{1}\mathbf{1}^\top / N$ by an arbitrary stochastic
+:math:`1 - \alpha` above.  Replacing :math:`\mathbf{1}\mathbf{1}^\top / N` by an arbitrary stochastic
 matrix produces the full DeGroot model and is a one-liner extension on the Rust side.
 
 Why it matters
@@ -66,7 +66,7 @@ Why it matters
   (Bayesian persuasion, social media echo chambers, voting-system stability).
 * **Distributed estimation & federated learning.**  Average-consensus protocols for sensor
   networks, gossip algorithms, federated averaging — all reduce to the same contraction
-  argument with explicit convergence rate $1 - \alpha$.
+  argument with explicit convergence rate :math:`1 - \alpha`.
 * **Coupled-oscillator physics.**  Linear approximation of the Kuramoto / Vicsek models near
   the synchronised regime; direct comparison with the McKean–Vlasov continuous limit.
 

docs/source/algorithms/bsde.rst

@@ -1,16 +1,16 @@
 BSDE — θ-scheme and deep-BSDE bridge
 ====================================
 
-A **backward stochastic differential equation** (BSDE) on $[0, T]$ is the inverse-time problem
+A **backward stochastic differential equation** (BSDE) on :math:`[0, T]` is the inverse-time problem
 
 .. math::
 
    Y_t \;=\; \xi \;+\; \int_t^T f(s, Y_s, Z_s)\, ds \;-\; \int_t^T Z_s\, dW_s,
    \qquad Y_T = \xi,
 
-where $\xi \in L^2(\mathcal{F}_T)$ is the *terminal condition*, $f$ is the *driver* and the
+where :math:`\xi \in L^2(\mathcal{F}_T)` is the *terminal condition*, :math:`f` is the *driver* and the
-unknowns are an adapted pair $(Y, Z) \in \mathcal{S}^2 \times \mathcal{H}^2$.  The auxiliary
+unknowns are an adapted pair :math:`(Y, Z) \in \mathcal{S}^2 \times \mathcal{H}^2`.  The auxiliary
-process $Z$ is a *non-anticipative hedge*: it makes the equation adapted despite the terminal
+process :math:`Z` is a *non-anticipative hedge*: it makes the equation adapted despite the terminal
 constraint.
 
 The primitive `linear_bsde_constant_coeffs` solves the constant-coefficient linear case
@@ -20,19 +20,19 @@ The primitive `linear_bsde_constant_coeffs` solves the constant-coefficient line
    -dY_t \;=\; (a\, Y_t + b\, Z_t + c)\, dt \;-\; Z_t\, dW_t,
    \qquad Y_T = \xi,
 
-by a **Crank–Nicolson θ-scheme** (θ = 0.5 → second-order in $\Delta t$).
+by a **Crank–Nicolson θ-scheme** (θ = 0.5 → second-order in :math:`\Delta t`).
 
 Mathematical background
 -----------------------
 
-**Pardoux–Peng theorem (1990).**  If $f$ is uniformly Lipschitz in $(y, z)$ and
+**Pardoux–Peng theorem (1990).**  If :math:`f` is uniformly Lipschitz in :math:`(y, z)` and
-$\mathbb{E}\!\int_0^T f(s, 0, 0)^2\, ds < \infty$, then the BSDE admits a unique solution
+:math:`\mathbb{E}\!\int_0^T f(s, 0, 0)^2\, ds < \infty`, then the BSDE admits a unique solution
-$(Y, Z) \in \mathcal{S}^2 \times \mathcal{H}^2$.  The proof is a Banach–Picard fixed point on
+:math:`(Y, Z) \in \mathcal{S}^2 \times \mathcal{H}^2`.  The proof is a Banach–Picard fixed point on
-$\Phi : (Y, Z) \mapsto (Y', Z')$ with
+:math:`\Phi : (Y, Z) \mapsto (Y', Z')` with
-$Y'_t = \mathbb{E}\bigl[\xi + \int_t^T f(s, Y_s, Z_s)\, ds \bigm| \mathcal{F}_t\bigr]$ and $Z'$
+:math:`Y'_t = \mathbb{E}\bigl[\xi + \int_t^T f(s, Y_s, Z_s)\, ds \bigm| \mathcal{F}_t\bigr]` and :math:`Z'`
 obtained by the martingale representation theorem.
 
-**Closed-form for the linear case.**  For $a, b, c$ deterministic the solution is the
+**Closed-form for the linear case.**  For :math:`a, b, c` deterministic the solution is the
 conditional expectation under a Girsanov-shifted measure:
 
 .. math::
@@ -41,12 +41,12 @@ conditional expectation under a Girsanov-shifted measure:
               \;+\; \int_t^T c(s)\, e^{\int_t^s a(r)\, dr}\, ds
               \,\Big|\, \mathcal{F}_t \right],
 
-with the Girsanov density $\frac{d\mathbb{Q}}{d\mathbb{P}} = \mathcal{E}\bigl(\int_0^\cdot b(s)\,dW_s\bigr)$.
+with the Girsanov density :math:`\frac{d\mathbb{Q}}{d\mathbb{P}} = \mathcal{E}\bigl(\int_0^\cdot b(s)\,dW_s\bigr)`.
-When $b = c = 0$, $a \equiv -\rho$ and $\xi = 1$ this collapses to the analytic ground truth
+When :math:`b = c = 0`, :math:`a \equiv -\rho` and :math:`\xi = 1` this collapses to the analytic ground truth
-$Y_t = e^{-\rho(T-t)}$ used by the convergence test.
+:math:`Y_t = e^{-\rho(T-t)}` used by the convergence test.
 
-**Feynman–Kac bridge.**  Setting $f(s, y, z) = -r y$ and $\xi = g(X_T)$ for a forward SDE $X$
+**Feynman–Kac bridge.**  Setting :math:`f(s, y, z) = -r y` and :math:`\xi = g(X_T)` for a forward SDE :math:`X`
-recovers the discounted-payoff PDE: $Y_t = e^{-r(T-t)} \mathbb{E}[g(X_T) \mid \mathcal{F}_t]$.
+recovers the discounted-payoff PDE: :math:`Y_t = e^{-r(T-t)} \mathbb{E}[g(X_T) \mid \mathcal{F}_t]`.
 More generally, the markovian BSDE
 
 .. math::
@@ -54,10 +54,10 @@ More generally, the markovian BSDE
    Y_t = g(X_T) + \int_t^T f(s, X_s, Y_s, Z_s)\, ds - \int_t^T Z_s\, dW_s,
 
 is the probabilistic representation of the semilinear PDE
-$\partial_t u + \mathcal{L}u + f(t, x, u, \sigma^\top \nabla u) = 0$, $u(T, x) = g(x)$, with
+:math:`\partial_t u + \mathcal{L}u + f(t, x, u, \sigma^\top \nabla u) = 0`, :math:`u(T, x) = g(x)`, with
-$Y_t = u(t, X_t)$ and $Z_t = \sigma^\top(t, X_t)\nabla u(t, X_t)$.
+:math:`Y_t = u(t, X_t)` and :math:`Z_t = \sigma^\top(t, X_t)\nabla u(t, X_t)`.
 
-**Crank–Nicolson θ-scheme.**  On a uniform grid $0 = t_0 < \cdots < t_N = T$ the scheme reads
+**Crank–Nicolson θ-scheme.**  On a uniform grid :math:`0 = t_0 < \cdots < t_N = T` the scheme reads
 
 .. math::
 
@@ -65,28 +65,28 @@ $Y_t = u(t, X_t)$ and $Z_t = \sigma^\top(t, X_t)\nabla u(t, X_t)$.
               \;+\; \Delta t\,\bigl(\theta\, f(t_i, Y^N_{t_i}, Z^N_{t_i})
                                      + (1-\theta)\, f(t_{i+1}, Y^N_{t_{i+1}}, Z^N_{t_{i+1}})\bigr),
 
-with $Z^N_{t_i} = \Delta t^{-1}\,\mathbb{E}\bigl[Y^N_{t_{i+1}}(W_{t_{i+1}} - W_{t_i})\bigm|\mathcal{F}_{t_i}\bigr]$
+with :math:`Z^N_{t_i} = \Delta t^{-1}\,\mathbb{E}\bigl[Y^N_{t_{i+1}}(W_{t_{i+1}} - W_{t_i})\bigm|\mathcal{F}_{t_i}\bigr]`
-(discrete Clark–Ocone identity).  For $\theta = 1/2$ the global truncation error is
+(discrete Clark–Ocone identity).  For :math:`\theta = 1/2` the global truncation error is
-$\sup_i \mathbb{E}|Y_{t_i} - Y^N_{t_i}|^2 = O(\Delta t^2)$ — the second-order rate verified
+:math:`\sup_i \mathbb{E}|Y_{t_i} - Y^N_{t_i}|^2 = O(\Delta t^2)` — the second-order rate verified
 empirically by the convergence cell of the companion notebook.
 
 **Deep-BSDE bridge (E–Han–Jentzen, 2017).**  In high dimension the conditional expectation
-is intractable; one parametrises $Z_{t_i} = \zeta^i_\theta(X_{t_i})$ by a neural network and
+is intractable; one parametrises :math:`Z_{t_i} = \zeta^i_\theta(X_{t_i})` by a neural network and
-minimises $\mathbb{E}\bigl[(Y^\theta_T - \xi)^2\bigr]$ over $(Y_0, \theta)$.  The trait
+minimises :math:`\mathbb{E}\bigl[(Y^\theta_T - \xi)^2\bigr]` over :math:`(Y_0, \theta)`.  The trait
 `ConditionalExpectation` and the struct `DeepBsdeBridge` expose the same θ-scheme step so the
 user can plug in any regression / neural-network conditional-expectation oracle.
 
 Why it matters
 --------------
 
-* **Pricing & hedging in incomplete markets.**  $Y_t$ is the super-replication price of the
+* **Pricing & hedging in incomplete markets.**  :math:`Y_t` is the super-replication price of the
-  contingent claim $\xi$ and $Z_t$ is the instantaneous hedge ratio.  Constraints (transaction
+  contingent claim :math:`\xi` and :math:`Z_t` is the instantaneous hedge ratio.  Constraints (transaction
-  costs, portfolio caps, recursive utilities) are absorbed into the driver $f$.
+  costs, portfolio caps, recursive utilities) are absorbed into the driver :math:`f`.
 * **Stochastic control.**  Forward–backward SDEs are the probabilistic counterpart of the
-  Hamilton–Jacobi–Bellman PDE; deep-BSDE solves HJB up to $d \sim 100$ state variables, well
+  Hamilton–Jacobi–Bellman PDE; deep-BSDE solves HJB up to :math:`d \sim 100` state variables, well
   beyond grid-based PDE solvers.
 * **Risk-sensitive optimisation.**  Quadratic-driver BSDE
-  $-dY = \tfrac1{2\eta}|Z|^2 dt - Z\, dW$ encodes exponential utility hedging (Kramkov–Schachermayer 1999).
+  :math:`-dY = \tfrac1{2\eta}|Z|^2 dt - Z\, dW` encodes exponential utility hedging (Kramkov–Schachermayer 1999).
 
 .. note::
    📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/10_bsde.ipynb>`_
@@ -108,7 +108,7 @@ Generic CPU-only Crank–Nicolson scheme for linear backward stochastic differen
 Exponential ground-truth check
 ------------------------------
 
-With $a(t) \equiv -\rho$, $b = c = 0$ and $Y_T = 1$ the analytic deterministic solution is $Y_t = e^{-\rho (T-t)}$.
+With :math:`a(t) \equiv -\rho`, :math:`b = c = 0` and :math:`Y_T = 1` the analytic deterministic solution is :math:`Y_t = e^{-\rho (T-t)}`.
 
 .. code-block:: python
 

docs/source/algorithms/generative_calibration_hooks.rst

@@ -8,9 +8,9 @@ function of every generative-calibration loop in `optimiz-rs`.
 Mathematical background
 -----------------------
 
-**Definition.**  For a positive-definite kernel $k : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$
+**Definition.**  For a positive-definite kernel :math:`k : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}`
-with reproducing-kernel Hilbert space (RKHS) $\mathcal{H}_k$, the *kernel mean embedding* of a
+with reproducing-kernel Hilbert space (RKHS) :math:`\mathcal{H}_k`, the *kernel mean embedding* of a
-probability measure $P$ is $\mu_P := \mathbb{E}_{X \sim P}[k(X, \cdot)] \in \mathcal{H}_k$.
+probability measure :math:`P` is :math:`\mu_P := \mathbb{E}_{X \sim P}[k(X, \cdot)] \in \mathcal{H}_k`.
 The **squared MMD** is the RKHS distance between embeddings:
 
 .. math::
@@ -18,10 +18,10 @@ The **squared MMD** is the RKHS distance between embeddings:
    \mathrm{MMD}^2(P, Q) \;:=\; \| \mu_P - \mu_Q \|_{\mathcal{H}_k}^2
      \;=\; \mathbb{E}\,[k(X, X')] \;-\; 2\, \mathbb{E}\,[k(X, Y)] \;+\; \mathbb{E}\,[k(Y, Y')] ,
 
-where $X, X' \sim P$ and $Y, Y' \sim Q$ are independent.  When $k$ is *characteristic*
+where :math:`X, X' \sim P` and :math:`Y, Y' \sim Q` are independent.  When :math:`k` is *characteristic*
-(e.g. Gaussian RBF), $\mathrm{MMD}(P, Q) = 0 \iff P = Q$.
+(e.g. Gaussian RBF), :math:`\mathrm{MMD}(P, Q) = 0 \iff P = Q`.
 
-**U-statistic estimator.**  Given i.i.d. samples $\{x_i\}_{i=1}^n$ and $\{y_j\}_{j=1}^m$, the
+**U-statistic estimator.**  Given i.i.d. samples :math:`\{x_i\}_{i=1}^n` and :math:`\{y_j\}_{j=1}^m`, the
 unbiased estimator is
 
 .. math::
@@ -31,21 +31,21 @@ unbiased estimator is
      \;-\; \frac{2}{n m}\!\sum_{i, j} k(x_i, y_j)
      \;+\; \frac{1}{m(m-1)}\!\sum_{j \ne j'} k(y_j, y_{j'}) .
 
-It is unbiased, computable in $O((n + m)^2)$ for $d = 1$ (the case implemented), and asymptotically
+It is unbiased, computable in :math:`O((n + m)^2)` for :math:`d = 1` (the case implemented), and asymptotically
 normal under the alternative.  Self-distance is **exactly zero**.
 
 **Kernel.**  The shipped routine uses the Gaussian RBF
-$k_\sigma(x, y) = \exp\!\bigl(-(x - y)^2 / (2\sigma^2)\bigr)$ with bandwidth $\sigma$.  Standard
+:math:`k_\sigma(x, y) = \exp\!\bigl(-(x - y)^2 / (2\sigma^2)\bigr)` with bandwidth :math:`\sigma`.  Standard
 reproducing-kernel theory shows that this kernel is *characteristic*, hence MMD metrises weak
 convergence on bounded subsets.
 
 **Closed forms for two notable cases.**
 
-* **Pure translation, equal samples.**  If $Q$ is the law of $X + \Delta$ with $X \sim P$ on
+* **Pure translation, equal samples.**  If :math:`Q` is the law of :math:`X + \Delta` with :math:`X \sim P` on
-  $\mathbb{R}$ and $P = \delta$ atomic, the squared MMD is $2 - 2 e^{-\Delta^2 / (2\sigma^2)}$ —
+  :math:`\mathbb{R}` and :math:`P = \delta` atomic, the squared MMD is :math:`2 - 2 e^{-\Delta^2 / (2\sigma^2)}` —
-  smooth, monotone in $|\Delta|$, asymptote $2$ as $\Delta \to \infty$.  This is the analytic
+  smooth, monotone in :math:`|\Delta|`, asymptote :math:`2` as :math:`\Delta \to \infty`.  This is the analytic
   ground-truth verified by the *bandwidth dependence* cell of the companion notebook.
-* **Two Gaussians.**  For $P = \mathcal{N}(\mu_1, \sigma_1^2)$ and $Q = \mathcal{N}(\mu_2, \sigma_2^2)$,
+* **Two Gaussians.**  For :math:`P = \mathcal{N}(\mu_1, \sigma_1^2)` and :math:`Q = \mathcal{N}(\mu_2, \sigma_2^2)`,
 
   .. math::
 
@@ -57,8 +57,8 @@ convergence on bounded subsets.
   giving an exact reference for unit tests.
 
 **Statistical guarantee.**  Gretton et al. (2012, Thm. 12) give the deviation bound
-$\Pr\!\bigl(\widehat{\mathrm{MMD}}^2 - \mathrm{MMD}^2 > \varepsilon\bigr) \le \exp\bigl(-\varepsilon^2 nm / (8 K^2 (n + m))\bigr)$
+:math:`\Pr\!\bigl(\widehat{\mathrm{MMD}}^2 - \mathrm{MMD}^2 > \varepsilon\bigr) \le \exp\bigl(-\varepsilon^2 nm / (8 K^2 (n + m))\bigr)`
-for $|k| \le K$.  Hence MMD detects fixed alternatives at the optimal $n^{-1/2}$ rate.
+for :math:`|k| \le K`.  Hence MMD detects fixed alternatives at the optimal :math:`n^{-1/2}` rate.
 
 **Connection with Wasserstein.**  Both metrise weak convergence, but MMD is *quadratic in the
 sample size* (no transport plan to solve) and admits unbiased low-variance gradient estimators —
@@ -69,7 +69,7 @@ Why it matters
 --------------
 
 * **Generative calibration.**  Train an implicit sampler (neural SDE, copula generator, GAN-like
-  architecture) by minimising $\widehat{\mathrm{MMD}}^2$ between the simulator output and the
+  architecture) by minimising :math:`\widehat{\mathrm{MMD}}^2` between the simulator output and the
   target distribution.  The trait `GenerativeSampler` plus `calibration_step` is the abstract
   glue.
 * **Two-sample testing.**  Distribution drift detection in streaming data, A/B-test signal

docs/source/algorithms/mckean_vlasov.rst

@@ -9,7 +9,7 @@ on the *law* of the solution itself:
    dX_t \;=\; b\bigl(t, X_t, \mathcal{L}(X_t)\bigr)\, dt \;+\; \sigma\bigl(t, X_t, \mathcal{L}(X_t)\bigr)\, dW_t,
    \qquad X_0 \sim \mu_0 .
 
-It is the formal $N \to \infty$ limit of an exchangeable system of $N$ interacting diffusions
+It is the formal :math:`N \to \infty` limit of an exchangeable system of :math:`N` interacting diffusions
 
 .. math::
 
@@ -23,22 +23,22 @@ The primitive shipped here, `mean_reverting_mckean_vlasov`, simulates the canoni
    dX_t \;=\; \theta\bigl(\bar X_t - X_t\bigr)\, dt \;+\; \sigma\, dW_t,
    \qquad \bar X_t = \mathbb{E}[X_t],
 
-with the symmetric Euler particle scheme $X^{i,N}_{k+1} = X^{i,N}_k + \theta(\bar X^N_k - X^{i,N}_k)\Delta t + \sigma\sqrt{\Delta t}\,\xi^i_k$.
+with the symmetric Euler particle scheme :math:`X^{i,N}_{k+1} = X^{i,N}_k + \theta(\bar X^N_k - X^{i,N}_k)\Delta t + \sigma\sqrt{\Delta t}\,\xi^i_k`.
 
 Mathematical background
 -----------------------
 
-**Sznitman's propagation of chaos (1991).**  Under standard Lipschitz assumptions on $b, \sigma$ in
+**Sznitman's propagation of chaos (1991).**  Under standard Lipschitz assumptions on :math:`b, \sigma` in
-$(x, \mu)$ (the $\mu$ argument equipped with the Wasserstein distance $W_2$), the empirical
+:math:`(x, \mu)` (the :math:`\mu` argument equipped with the Wasserstein distance :math:`W_2`), the empirical
-measure $\mu^N_t = \tfrac1N \sum_i \delta_{X^{i,N}_t}$ converges weakly to the deterministic flow
+measure :math:`\mu^N_t = \tfrac1N \sum_i \delta_{X^{i,N}_t}` converges weakly to the deterministic flow
-$\mathcal{L}(X_t)$, and any fixed sub-system of $k$ particles becomes asymptotically independent:
+:math:`\mathcal{L}(X_t)`, and any fixed sub-system of :math:`k` particles becomes asymptotically independent:
 
 .. math::
 
    \sup_{0 \le t \le T} \, \mathbb{E}\bigl[\,W_2^2\!\bigl(\mu^N_t,\, \mathcal{L}(X_t)\bigr)\bigr]
    \;\le\; \frac{C(T)}{N^{2/(d+4)}} .
 
-**Density flow (nonlinear Fokker–Planck).**  The marginal density $\rho_t = \mathrm{law}(X_t)$
+**Density flow (nonlinear Fokker–Planck).**  The marginal density :math:`\rho_t = \mathrm{law}(X_t)`
 satisfies the *nonlinear* PDE
 
 .. math::
@@ -47,8 +47,8 @@ satisfies the *nonlinear* PDE
    \;=\; \tfrac12\, \nabla^2\!:\!\bigl(\sigma\sigma^\top(t, x, \rho_t)\, \rho_t\bigr).
 
 **Closed-form for the mean-reverting case.**  Taking expectation of the SDE gives
-$\dot{\bar X}_t = 0$, so the population mean is *exactly preserved*: $\bar X_t \equiv \bar X_0$.
+:math:`\dot{\bar X}_t = 0`, so the population mean is *exactly preserved*: :math:`\bar X_t \equiv \bar X_0`.
-The deviation $\widetilde X^i_t := X^{i,N}_t - \bar X_0$ then solves a standard Ornstein–Uhlenbeck
+The deviation :math:`\widetilde X^i_t := X^{i,N}_t - \bar X_0` then solves a standard Ornstein–Uhlenbeck
 SDE, so each marginal is Gaussian with
 
 .. math::
@@ -61,21 +61,21 @@ SDE, so each marginal is Gaussian with
 The companion notebook checks both the mean conservation and the variance asymptote.
 
 **Connection with mean-field BSDEs.**  Coupling the McKean–Vlasov forward SDE with a backward
-equation $-dY_t = f(t, X_t, Y_t, Z_t, \mathcal{L}(X_t, Y_t))\, dt - Z_t\, dW_t$ produces the
+equation :math:`-dY_t = f(t, X_t, Y_t, Z_t, \mathcal{L}(X_t, Y_t))\, dt - Z_t\, dW_t` produces the
 *mean-field BSDE* of Carmona–Delarue (2018), itself the probabilistic representation of the
 HJB side of mean-field games (cf. :doc:`stochastic_control`).
 
 Why it matters
 --------------
 
-* **Mean-field games.**  At the Nash equilibrium of a symmetric $N$-player game, each player's
+* **Mean-field games.**  At the Nash equilibrium of a symmetric :math:`N`-player game, each player's
-  state follows a McKean–Vlasov SDE in which the population law $\mu_t$ is the consistent
+  state follows a McKean–Vlasov SDE in which the population law :math:`\mu_t` is the consistent
   fixed point of every player's best response.  This is the master tool of Lasry–Lions theory
   for systemic-risk modelling, optimal execution and price formation.
 * **Statistical physics.**  Vlasov, Boltzmann, and granular-media equations all arise as
   density flows of mean-field particle systems; the same Euler scheme estimates their solutions.
 * **Generative modelling.**  Stein-variational gradient descent and score-based diffusion can
-  be analysed as McKean–Vlasov gradient flows on $W_2$.
+  be analysed as McKean–Vlasov gradient flows on :math:`W_2`.
 
 .. note::
    📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/14_mckean_vlasov.ipynb>`_

docs/source/algorithms/pde.rst

@@ -10,7 +10,7 @@ Mathematical background
 -----------------------
 
 **Fokker–Planck (Kolmogorov forward).**  For a 1-D Itô diffusion
-$dX_t = \mu(t, x)\, dt + \sigma(t, x)\, dW_t$, the marginal density $\rho(t, x)$ of $X_t$
+:math:`dX_t = \mu(t, x)\, dt + \sigma(t, x)\, dW_t`, the marginal density :math:`\rho(t, x)` of :math:`X_t`
 satisfies the parabolic PDE
 
 .. math::
@@ -19,15 +19,15 @@ satisfies the parabolic PDE
    \;=\; \tfrac12\, \partial^2_{xx}\!\bigl(\sigma^2(t,x)\, \rho\bigr),
    \qquad \rho(0, \cdot) = \rho_0 .
 
-For the *pure-diffusion* test ($\mu \equiv 0$, $\sigma^2 \equiv 1$, $\rho_0 = \mathcal{N}(0, 1)$)
+For the *pure-diffusion* test (:math:`\mu \equiv 0`, :math:`\sigma^2 \equiv 1`, :math:`\rho_0 = \mathcal{N}(0, 1)`)
-the analytic Gaussian heat kernel gives $\rho(t, x) = \frac{1}{\sqrt{2\pi(1+t)}}\exp\!\bigl(-\frac{x^2}{2(1+t)}\bigr)$,
+the analytic Gaussian heat kernel gives :math:`\rho(t, x) = \frac{1}{\sqrt{2\pi(1+t)}}\exp\!\bigl(-\frac{x^2}{2(1+t)}\bigr)`,
-so the variance grows linearly: $\mathrm{Var}(X_t) = 1 + t$.  The conservative
+so the variance grows linearly: :math:`\mathrm{Var}(X_t) = 1 + t`.  The conservative
 Lax–Wendroff / centred-flux scheme implemented by `fokker_planck_constant` preserves total mass
 (checked in the notebook to machine precision).
 
 **Hamilton–Jacobi–Bellman.**  Consider the controlled diffusion
-$dX_t = \mu(X_t, \alpha_t)\, dt + \sigma(X_t)\, dW_t$ and the value function
+:math:`dX_t = \mu(X_t, \alpha_t)\, dt + \sigma(X_t)\, dW_t` and the value function
-$v(t, x) = \sup_\alpha \mathbb{E}_{t,x}\!\bigl[\int_t^T r(X_s, \alpha_s)\, ds + g(X_T)\bigr]$.
+:math:`v(t, x) = \sup_\alpha \mathbb{E}_{t,x}\!\bigl[\int_t^T r(X_s, \alpha_s)\, ds + g(X_T)\bigr]`.
 Dynamic programming produces
 
 .. math::
@@ -41,24 +41,24 @@ Dynamic programming produces
 heat-only relaxation case (:math:`H \equiv 0`, :math:`\sigma^2 > 0`) preserves a constant value while a
 quadratic terminal :math:`g(x) = \tfrac12 \lVert x \rVert^2` smooths into a Gaussian-shaped value surface.
 
-**Elliptic Poisson with zero Dirichlet boundary.**  On the unit square $\Omega = (0,1)^2$,
+**Elliptic Poisson with zero Dirichlet boundary.**  On the unit square :math:`\Omega = (0,1)^2`,
 
 .. math::
 
    -\Delta u(x, y) = f(x, y) \text{ in } \Omega, \qquad u\!\restriction_{\partial\Omega} = 0 .
 
-The Laplace eigenfunctions $\phi_{m,n}(x, y) = \sin(m\pi x)\sin(n\pi y)$ form an
+The Laplace eigenfunctions :math:`\phi_{m,n}(x, y) = \sin(m\pi x)\sin(n\pi y)` form an
-orthonormal basis with eigenvalues $\lambda_{m,n} = (m^2 + n^2)\pi^2$, so for
+orthonormal basis with eigenvalues :math:`\lambda_{m,n} = (m^2 + n^2)\pi^2`, so for
-$f = 2\pi^2 \sin(\pi x)\sin(\pi y)$ the *exact* solution is
+:math:`f = 2\pi^2 \sin(\pi x)\sin(\pi y)` the *exact* solution is
-$u(x, y) = \sin(\pi x)\sin(\pi y)$.  `poisson_2d_zero_boundary` solves the 5-point stencil by
+:math:`u(x, y) = \sin(\pi x)\sin(\pi y)`.  `poisson_2d_zero_boundary` solves the 5-point stencil by
 **Successive Over-Relaxation** with optimal relaxation parameter
-$\omega^* = 2 / (1 + \sin(\pi h))$ for grid spacing $h = 1/(N-1)$, achieving spectral radius
+:math:`\omega^* = 2 / (1 + \sin(\pi h))` for grid spacing :math:`h = 1/(N-1)`, achieving spectral radius
-$\rho \sim 1 - 2\pi h$ — i.e. $O(h^{-1})$ iterations to reach a fixed tolerance, against
+:math:`\rho \sim 1 - 2\pi h` — i.e. :math:`O(h^{-1})` iterations to reach a fixed tolerance, against
-$O(h^{-2})$ for plain Gauss–Seidel.
+:math:`O(h^{-2})` for plain Gauss–Seidel.
 
 **Probabilistic representation (Feynman–Kac).**  Both the parabolic HJB and the elliptic
-Poisson PDE admit stochastic representations: $u(x) = \mathbb{E}_x\!\bigl[\int_0^{\tau_\Omega} f(X_s)\, ds\bigr]$
+Poisson PDE admit stochastic representations: :math:`u(x) = \mathbb{E}_x\!\bigl[\int_0^{\tau_\Omega} f(X_s)\, ds\bigr]`
-for the latter, where $\tau_\Omega$ is the first exit time of the diffusion from $\Omega$.
+for the latter, where :math:`\tau_\Omega` is the first exit time of the diffusion from :math:`\Omega`.
 This links the PDE solvers above to the BSDE primitives of :doc:`bsde`.
 
 Why it matters
@@ -95,7 +95,7 @@ Fokker–Planck, HJB, Poisson.
 Pure-diffusion Fokker–Planck
 ----------------------------
 
-$\partial_t m = \tfrac12 \partial_{xx} m$ with Gaussian initial density should remain centred and approximately Gaussian.
+:math:`\partial_t m = \tfrac12 \partial_{xx} m` with Gaussian initial density should remain centred and approximately Gaussian.
 
 .. code-block:: python
 
@@ -140,7 +140,7 @@ $\partial_t m = \tfrac12 \partial_{xx} m$ with Gaussian initial density should r
 2-D Poisson eigenfunction
 -------------------------
 
-$-\Delta u = 2\pi^2 \sin(\pi x)\sin(\pi y)$ on the unit square with zero Dirichlet boundary admits the exact solution $u(x,y) = \sin(\pi x)\sin(\pi y)$.
+:math:`-\Delta u = 2\pi^2 \sin(\pi x)\sin(\pi y)` on the unit square with zero Dirichlet boundary admits the exact solution :math:`u(x,y) = \sin(\pi x)\sin(\pi y)`.
 
 .. code-block:: python
 
@@ -183,7 +183,7 @@ $-\Delta u = 2\pi^2 \sin(\pi x)\sin(\pi y)$ on the unit square with zero Dirichl
 2-D HJB with quadratic terminal
 -------------------------------
 
-Heat-only relaxation ($H = 0$, σ² > 0) preserves a constant value, while a quadratic terminal $g(x) = ½(x²+y²)$ smooths.
+Heat-only relaxation (:math:`H = 0`, σ² > 0) preserves a constant value, while a quadratic terminal :math:`g(x) = ½(x²+y²)` smooths.
 
 .. code-block:: python
 

docs/source/algorithms/quadratic_impact_control.rst

@@ -7,7 +7,7 @@ problem with running quadratic *impact* penalty.
 Mathematical background
 -----------------------
 
-Let $A_t$ be a controlled scalar state driven by an additive control $u_t$ and Gaussian noise.
+Let :math:`A_t` be a controlled scalar state driven by an additive control :math:`u_t` and Gaussian noise.
 The controller minimises the *finite-horizon quadratic objective*
 
 .. math::
@@ -16,11 +16,11 @@ The controller minimises the *finite-horizon quadratic objective*
      \;+\; \tfrac{\phi}{2}\, A_t^2 \,\bigr)\, dt
      \;+\; \tfrac{A_T}{2}\, A_T^2 \,\right] ,
 
-where $\gamma > 0$ is the **impact / control cost**, $\phi \ge 0$ the **running risk weight**
+where :math:`\gamma > 0` is the **impact / control cost**, :math:`\phi \ge 0` the **running risk weight**
-and $A_T$ the **terminal penalty** (over-loaded notation: $A_T$ here is the *coefficient*).
+and :math:`A_T` the **terminal penalty** (over-loaded notation: :math:`A_T` here is the *coefficient*).
 
-**Hamilton–Jacobi–Bellman.**  With value function $v(t, A) = \tfrac12 h(t)\, A^2 + c(t)$, the
+**Hamilton–Jacobi–Bellman.**  With value function :math:`v(t, A) = \tfrac12 h(t)\, A^2 + c(t)`, the
-HJB equation collapses to a scalar Riccati ODE on $h$:
+HJB equation collapses to a scalar Riccati ODE on :math:`h`:
 
 .. math::
 
@@ -34,15 +34,15 @@ The optimal feedback is the linear law
 
    u^*(t, A) \;=\; -\, \frac{h(t)}{\gamma}\, A \;\equiv\; -\, k(t)\, A,
 
-with *feedback gain* $k(t) = h(t) / \gamma$.  This is the structure returned by the primitive.
+with *feedback gain* :math:`k(t) = h(t) / \gamma`.  This is the structure returned by the primitive.
 
 **Closed-form solutions.**
 
-* **Symmetric fixed point** $\gamma = \phi = A_T = 1$: $h(t) \equiv 1$ is the unique solution
+* **Symmetric fixed point** :math:`\gamma = \phi = A_T = 1`: :math:`h(t) \equiv 1` is the unique solution
-  (RHS vanishes), so the feedback gain is constant $k \equiv 1$.  The notebook checks this
+  (RHS vanishes), so the feedback gain is constant :math:`k \equiv 1`.  The notebook checks this
   to machine precision.
-* **Generic $\phi > 0$.**  Writing $\bar h = \sqrt{\gamma \phi}$ for the steady-state and
+* **Generic :math:`\phi > 0`.**  Writing :math:`\bar h = \sqrt{\gamma \phi}` for the steady-state and
-  $\rho = \sqrt{\phi / \gamma}$, the Riccati ODE has the closed-form (separation of variables /
+  :math:`\rho = \sqrt{\phi / \gamma}`, the Riccati ODE has the closed-form (separation of variables /
   Bernoulli substitution)
 
   .. math::
@@ -50,19 +50,19 @@ with *feedback gain* $k(t) = h(t) / \gamma$.  This is the structure returned by
      h(t) \;=\; \bar h\, \frac{(\bar h + A_T)\, e^{2\rho(T-t)} \;-\; (\bar h - A_T)}
                               {(\bar h + A_T)\, e^{2\rho(T-t)} \;+\; (\bar h - A_T)} .
 
-  In the limit $T - t \to \infty$ the trajectory relaxes to the stationary value $\bar h = \sqrt{\gamma\phi}$.
+  In the limit :math:`T - t \to \infty` the trajectory relaxes to the stationary value :math:`\bar h = \sqrt{\gamma\phi}`.
-* **Free of running risk** $\phi = 0$.  Then $h'(t) = h(t)^2/\gamma$ integrates explicitly to
+* **Free of running risk** :math:`\phi = 0`.  Then :math:`h'(t) = h(t)^2/\gamma` integrates explicitly to
 
   .. math::
 
      h(t) \;=\; \frac{A_T}{1 + (A_T / \gamma)(T - t)} ,
 
-  recovering the Pontryagin LQR closed form $P(0) = 1/2$ of :doc:`stochastic_control`.
+  recovering the Pontryagin LQR closed form :math:`P(0) = 1/2` of :doc:`stochastic_control`.
 
 **Connection with mean-field games.**  Coupling this single-agent control with an interacting
-population — the running cost depending on the *average* control $\bar u_t$ — yields the
+population — the running cost depending on the *average* control :math:`\bar u_t` — yields the
 Almgren–Chriss MFG (Lasry–Lions 2007); at the Nash equilibrium the optimal trajectory is the
-uniform schedule $\dot A^*_t = -A_0 / T$ (cf. Sec. 3 of Carmona–Delarue 2018, Vol. I).
+uniform schedule :math:`\dot A^*_t = -A_0 / T` (cf. Sec. 3 of Carmona–Delarue 2018, Vol. I).
 
 Why it matters
 --------------
@@ -71,7 +71,7 @@ Why it matters
   Riccati ODE; the closed form means *real-time* feedback re-computation.
 * **Stochastic regulators.**  Temperature stabilisation, attitude control, queueing-network
   smoothing all map to a quadratic-impact problem with a single state.
-* **Building block for higher-dimensional MPC.**  Vector generalisations of $h(t)$ are matrix
+* **Building block for higher-dimensional MPC.**  Vector generalisations of :math:`h(t)` are matrix
   Riccati ODEs; this scalar primitive is the verification kernel against which the matrix
   solver in :doc:`matrix_riccati` is tested.
 
@@ -93,7 +93,7 @@ Why it matters
 Riccati fixed-point check
 -------------------------
 
-$h'(t) = h(t)^2/γ - φ$ with $h(T) = A$.  When $γ = φ = A = 1$ the right-hand side is $h^2 - 1 = 0$ at $h = 1$, so `h ≡ 1`.
+:math:`h'(t) = h(t)^2/γ - φ` with :math:`h(T) = A`.  When :math:`γ = φ = A = 1` the right-hand side is :math:`h^2 - 1 = 0` at :math:`h = 1`, so `h ≡ 1`.
 
 .. code-block:: python
 
@@ -134,7 +134,7 @@ $h'(t) = h(t)^2/γ - φ$ with $h(T) = A$.  When $γ = φ = A = 1$ the right-hand
 Sensitivity to the terminal weight
 ----------------------------------
 
-Vary $A$, fix $γ = 1$, $φ = 0.25$, $T = 1$.
+Vary :math:`A`, fix :math:`γ = 1`, :math:`φ = 0.25`, :math:`T = 1`.
 
 .. code-block:: python
 

docs/source/algorithms/robust_drift.rst

@@ -8,16 +8,16 @@ Heavy-tail-resistant maximum-likelihood estimator for the discrete Ornstein–Uh
    x_{k+1} \;=\; x_k \;+\; (a + b\, x_k)\, \Delta t \;+\; \sigma\, \sqrt{\Delta t}\, \varepsilon_k,
    \qquad \varepsilon_k \sim_{\text{i.i.d.}} P_\varepsilon ,
 
-where $P_\varepsilon$ is *contaminated*: a fraction $1 - \eta$ of standard Gaussian innovations
+where :math:`P_\varepsilon` is *contaminated*: a fraction :math:`1 - \eta` of standard Gaussian innovations
-plus a fraction $\eta$ of large outliers (jumps, fat tails, recording errors).
+plus a fraction :math:`\eta` of large outliers (jumps, fat tails, recording errors).
 
 Mathematical background
 -----------------------
 
-**Naive OLS.**  Setting $y_k := (x_{k+1} - x_k)/\Delta t$, the model is the linear regression
+**Naive OLS.**  Setting :math:`y_k := (x_{k+1} - x_k)/\Delta t`, the model is the linear regression
-$y_k = a + b\, x_k + \sigma\, \Delta t^{-1/2}\, \varepsilon_k$.  Ordinary least-squares
+:math:`y_k = a + b\, x_k + \sigma\, \Delta t^{-1/2}\, \varepsilon_k`.  Ordinary least-squares
-minimises $\sum_k (y_k - a - b x_k)^2$ but its breakdown point is $0$: a single outlier with
+minimises :math:`\sum_k (y_k - a - b x_k)^2` but its breakdown point is :math:`0`: a single outlier with
-$|\varepsilon_k| \gg 1$ moves the estimate arbitrarily far.
+:math:`|\varepsilon_k| \gg 1` moves the estimate arbitrarily far.
 
 **Huber loss & IRLS.**  Huber (1964) replaces the quadratic loss by the *piecewise* loss
 
@@ -30,7 +30,7 @@ $|\varepsilon_k| \gg 1$ moves the estimate arbitrarily far.
    \end{cases}
 
 which is *quadratic in the bulk* and *linear in the tails*.  The first-order condition
-$\sum_k \psi_\delta(r_k)\, \nabla_{a,b}\, r_k = 0$ with $\psi_\delta = \rho_\delta'$ rewrites
+:math:`\sum_k \psi_\delta(r_k)\, \nabla_{a,b}\, r_k = 0` with :math:`\psi_\delta = \rho_\delta'` rewrites
 as a weighted least-squares problem with weights
 
 .. math::
@@ -45,16 +45,16 @@ so the **Iteratively Reweighted Least-Squares** algorithm reads
    \qquad w^{(t+1)}_k = \min\!\bigl(1, \delta / |r^{(t+1)}_k|\bigr).
 
 The sequence converges geometrically when the design matrix is well-conditioned
-(Holland–Welsch 1977).  `robust_drift` returns the limit pair $(\widehat a, \widehat b)$ and
+(Holland–Welsch 1977).  `robust_drift` returns the limit pair :math:`(\widehat a, \widehat b)` and
 the number of iterations.
 
-**Choice of the cut-off.**  The default $\delta = 1.345 \cdot \hat\sigma$ delivers $95\%$
+**Choice of the cut-off.**  The default :math:`\delta = 1.345 \cdot \hat\sigma` delivers :math:`95\%`
 asymptotic efficiency under Gaussian innovations while keeping the influence function bounded;
 it is the Huber–Hampel value used as the standard reference in robust statistics.
 
 **Closed-form one-step (debiased OLS).**  When the contamination is symmetric and the
-innovations have finite variance $\sigma^2_\varepsilon$, the *consistent* one-step estimate at
+innovations have finite variance :math:`\sigma^2_\varepsilon`, the *consistent* one-step estimate at
-the ordinary least-squares solution $(\hat a^0, \hat b^0)$ reads
+the ordinary least-squares solution :math:`(\hat a^0, \hat b^0)` reads
 
 .. math::
 
@@ -63,12 +63,12 @@ the ordinary least-squares solution $(\hat a^0, \hat b^0)$ reads
    \binom{\hat a^0}{\hat b^0}
    \;+\; \bigl(X^\top W X\bigr)^{-1}\, X^\top \psi_\delta(r^0),
 
-where $X$ is the $(N - 1) \times 2$ design matrix and $W = \mathrm{diag}(w_k)$.  Bahadur
+where :math:`X` is the :math:`(N - 1) \times 2` design matrix and :math:`W = \mathrm{diag}(w_k)`.  Bahadur
-linearisation shows $\widehat\theta - \theta^\star = O_P(N^{-1/2})$ even in the contaminated
+linearisation shows :math:`\widehat\theta - \theta^\star = O_P(N^{-1/2})` even in the contaminated
-model, with asymptotic variance $\sigma^2_\psi / I^2_\psi$ (Huber, *Robust Statistics*, 2004,
+model, with asymptotic variance :math:`\sigma^2_\psi / I^2_\psi` (Huber, *Robust Statistics*, 2004,
 Thm. 7.7).
 
-**Connection with Malliavin calculus.**  The driver $a + b\, x$ is exactly the linearised
+**Connection with Malliavin calculus.**  The driver :math:`a + b\, x` is exactly the linearised
 drift of the Ornstein–Uhlenbeck process used in the Greeks formulae of
 :doc:`stochastic_control` and the Vasicek interest-rate model; robust calibration is the
 pre-requisite for any Monte-Carlo Greeks computation under noisy historical data.
@@ -79,7 +79,7 @@ Why it matters
 * **Heavy-tailed historical data.**  Crypto returns, electricity prices, plasma confinement
   signals, and bio-medical recordings all contain spikes that destroy OLS but leave Huber
   estimates within statistical noise.
-* **Online & streaming estimation.**  IRLS with $\sim 10$ iterations is real-time on streaming
+* **Online & streaming estimation.**  IRLS with :math:`\sim 10` iterations is real-time on streaming
   windows and exposes a stable derivative for downstream control loops.
 * **Robust risk management.**  Replacing raw OLS by IRLS in any volatility / mean-reversion
   estimator dramatically reduces *parameter risk* in stress periods.

docs/source/algorithms/stochastic_control.rst

@@ -9,9 +9,9 @@ intensity controller** for jump processes.
 Mathematical background
 -----------------------
 
-**1. Optimal switching as a Snell envelope.**  Let $(Y^i_k)_{k, i}$ be the running rewards in
+**1. Optimal switching as a Snell envelope.**  Let :math:`(Y^i_k)_{k, i}` be the running rewards in
-mode $i \in \{1, \dots, M\}$ and $c_{ij}$ the cost of switching from $i$ to $j$.  The value
+mode :math:`i \in \{1, \dots, M\}` and :math:`c_{ij}` the cost of switching from :math:`i` to :math:`j`.  The value
-function $V_k(i)$ satisfies the backward dynamic-programming recursion
+function :math:`V_k(i)` satisfies the backward dynamic-programming recursion
 
 .. math::
 
@@ -20,22 +20,22 @@ function $V_k(i)$ satisfies the backward dynamic-programming recursion
    V_k(i) \;=\; Y^i_k \;+\; \max_{j}\!\bigl( V_{k+1}(j) - c_{ij}\bigr).
 
 This is the *multi-mode Snell envelope* of El Karoui–Quenez (1995).  When switching is free
-($c_{ij} = 0$) and only mode 1 pays a unit reward at every period, $V_k(i) = N - k$ for
+(:math:`c_{ij} = 0`) and only mode 1 pays a unit reward at every period, :math:`V_k(i) = N - k` for
-$i \neq 1$ and $V_k(1) = N - k + 1$ — reproduced exactly by `optimal_switching_dp`.
+:math:`i \neq 1` and :math:`V_k(1) = N - k + 1` — reproduced exactly by `optimal_switching_dp`.
 
 **2. Pontryagin–Bismut maximum principle (LQR).**  For the controlled SDE
-$dX_t = (a X_t + b u_t)\, dt + \sigma\, dW_t$ with quadratic cost
+:math:`dX_t = (a X_t + b u_t)\, dt + \sigma\, dW_t` with quadratic cost
-$J(u) = \mathbb{E}\!\bigl[\int_0^T (q X_t^2 + r u_t^2)\, dt + s_T X_T^2\bigr]$, the
+:math:`J(u) = \mathbb{E}\!\bigl[\int_0^T (q X_t^2 + r u_t^2)\, dt + s_T X_T^2\bigr]`, the
-adjoint variable $P_t$ solves the **matrix Riccati ODE**
+adjoint variable :math:`P_t` solves the **matrix Riccati ODE**
 
 .. math::
 
    \dot P_t \;+\; 2 a\, P_t \;-\; \frac{b^2}{r}\, P_t^2 \;+\; q \;=\; 0,
    \qquad P_T = s_T,
 
-and the optimal feedback is $u^*_t = -(b/r)\, P_t\, X_t$.  In the canonical case
+and the optimal feedback is :math:`u^*_t = -(b/r)\, P_t\, X_t`.  In the canonical case
-$a = q = 0$, $b = r = s_T = 1$, $T = 1$ the ODE simplifies to
+:math:`a = q = 0`, :math:`b = r = s_T = 1`, :math:`T = 1` the ODE simplifies to
-$\dot P_t = P_t^2$, whose closed-form solution is
+:math:`\dot P_t = P_t^2`, whose closed-form solution is
 
 .. math::
 
@@ -43,8 +43,8 @@ $\dot P_t = P_t^2$, whose closed-form solution is
    \qquad
    P(0) = \tfrac12 .
 
-The primitive `pontryagin_lqr` reproduces this with relative error below $10^{-3}$ for
+The primitive `pontryagin_lqr` reproduces this with relative error below :math:`10^{-3}` for
-$N = 2000$ steps (the symmetric Strang splitting is second-order in $\Delta t$).
+:math:`N = 2000` steps (the symmetric Strang splitting is second-order in :math:`\Delta t`).
 
 **3. Two-sided intensity control.**  For a jump-controller the agent picks the rates
 :math:`\lambda_\pm \ge 0` at which up/down events fire.  With *affine premia*
@@ -72,7 +72,7 @@ Why it matters
 * **Optimal switching** powers production-mode selection (start/stop a power plant), regime
   changes in algorithmic strategies, and American-style option pricing (Carmona–Touzi 2008).
 * **Pontryagin LQR** is the linearised core of every continuous-control problem: target
-  tracking, Kalman-LQG, ground-up RL, robust $H_\infty$ design.
+  tracking, Kalman-LQG, ground-up RL, robust :math:`H_\infty` design.
 * **Two-sided intensity control** is the closed-form heart of optimal market making
   (Avellaneda–Stoikov 2008, Cartea–Jaimungal–Penalva 2015) and limit-order placement.
 
@@ -136,7 +136,7 @@ Two modes; only mode 1 pays a unit reward.  Free switching should give `V_0(0) =
 Pontryagin 1-D LQR
 ------------------
 
-Closed-form Riccati for $a=q=0$, $b=r=s_T=1$, $T=1$ is $P(t) = 1/(1 + (T - t))$, hence $P(0) = 0.5$.
+Closed-form Riccati for :math:`a=q=0`, :math:`b=r=s_T=1`, :math:`T=1` is :math:`P(t) = 1/(1 + (T - t))`, hence :math:`P(0) = 0.5`.
 
 .. code-block:: python