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/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
-$(x, \mu)$ (the $\mu$ argument equipped with the Wasserstein distance $W_2$), the empirical
-measure $\mu^N_t = \tfrac1N \sum_i \delta_{X^{i,N}_t}$ converges weakly to the deterministic flow
-$\mathcal{L}(X_t)$, and any fixed sub-system of $k$ particles becomes asymptotically independent:
+**Sznitman's propagation of chaos (1991).**  Under standard Lipschitz assumptions on :math:`b, \sigma` in
+:math:`(x, \mu)` (the :math:`\mu` argument equipped with the Wasserstein distance :math:`W_2`), the empirical
+measure :math:`\mu^N_t = \tfrac1N \sum_i \delta_{X^{i,N}_t}` converges weakly to the deterministic flow
+: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*: $\bar X_t \equiv \bar X_0$.
-The deviation $\widetilde X^i_t := X^{i,N}_t - \bar X_0$ then solves a standard Ornstein–Uhlenbeck
+:math:`\dot{\bar X}_t = 0`, so the population mean is *exactly preserved*: :math:`\bar X_t \equiv \bar X_0`.
+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
-  state follows a McKean–Vlasov SDE in which the population law $\mu_t$ is the consistent
+* **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 :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>`_