docs(v2.0.0-alpha.5): rich math+physics background per chapter, fix notebook download links
Each of the eight v2.0 algorithm pages (bsde, pde, stochastic_control, quadratic_impact_control, mckean_vlasov, agent_based, robust_drift, generative_calibration_hooks) gains: - A dedicated 'Mathematical background' section with the central theorem (Pardoux-Peng, Sznitman propagation of chaos, Pontryagin-Bismut, Huber-IRLS, Gretton MMD, Kolmogorov forward, etc.), key derivations and the analytic closed-form solution that the unit tests target. - An 'Applications' / 'Why it matters' paragraph listing concrete research and engineering use-cases so newcomers grasp the value of each primitive. - A repaired companion-notebook block: the broken relative path '../../examples/notebooks/...ipynb' (which 404s on RTD) is replaced by an explicit GitHub blob (view) + raw (download) URL pair. Sphinx now builds the full doc set with zero new warnings.
This commit is contained in:
ThotDjehuty
@@ -1,9 +1,78 @@
|
||||
Agent-based — bounded-confidence consensus
|
||||
==========================================
|
||||
|
||||
Generic interacting-agent simulator (`consensus_dynamics`) — linear bounded-confidence rule $s_i^{k+1} = (1-α) s_i^k + α \bar s^k + ξ_i$.
|
||||
Generic symmetric *interacting-agent* simulator implementing the linear bounded-confidence
|
||||
update rule
|
||||
|
||||
.. note:: Companion executed notebook: `15_agent_based.ipynb <../../examples/notebooks/15_agent_based.ipynb>`_
|
||||
.. math::
|
||||
|
||||
s^{k+1}_i \;=\; (1 - \alpha)\, s^k_i \;+\; \alpha\, \bar s^k \;+\; \xi^k_i,
|
||||
\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
|
||||
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
|
||||
$\bar s^{k+1} = \bar s^k + \bar\xi^k$ with $\mathbb{E}[\bar\xi^k] = 0$, 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*.
|
||||
|
||||
**Geometric contraction of the spread.** Define the deviation $d^k_i := s^k_i - \bar s^k$.
|
||||
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
|
||||
*contracts geometrically* with rate $1 - \alpha$. 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
|
||||
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
|
||||
|
||||
.. 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 $\alpha = \theta\, \Delta t$,
|
||||
$\xi^k_i = \sigma \sqrt{\Delta t}\, W^i_k$ and $\Delta t \to 0$ 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
|
||||
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
|
||||
complete-graph case shipped here has $L = I - \tfrac1N \mathbf{1}\mathbf{1}^\top$ with
|
||||
eigenvalue $1$ on the orthogonal complement of $\mathbf{1}$, hence the contraction rate
|
||||
$1 - \alpha$ above. Replacing $\mathbf{1}\mathbf{1}^\top / N$ by an arbitrary stochastic
|
||||
matrix produces the full DeGroot model and is a one-liner extension on the Rust side.
|
||||
|
||||
Why it matters
|
||||
--------------
|
||||
|
||||
* **Opinion dynamics & social learning.** Calibration of polarisation/consensus models
|
||||
(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 $1 - \alpha$.
|
||||
* **Coupled-oscillator physics.** Linear approximation of the Kuramoto / Vicsek models near
|
||||
the synchronised regime; direct comparison with the McKean–Vlasov continuous limit.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/15_agent_based.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/15_agent_based.ipynb>`_
|
||||
|
||||
15 — Agent-based dynamics
|
||||
=========================
|
||||
|
||||
@@ -1,10 +1,96 @@
|
||||
BSDE — θ-scheme and deep-BSDE bridge
|
||||
====================================
|
||||
|
||||
This notebook exercises `optimizr.linear_bsde_constant_coeffs`, the Crank–Nicolson θ-scheme for the BSDE
|
||||
`-dY = (a Y + b Z + c) dt - Z dW` with constant coefficients, and verifies the discrete trajectory against the analytic solution `Y_t = exp(-ρ (T - t))`.
|
||||
A **backward stochastic differential equation** (BSDE) on $[0, T]$ is the inverse-time problem
|
||||
|
||||
.. note:: Companion executed notebook: `10_bsde.ipynb <../../examples/notebooks/10_bsde.ipynb>`_
|
||||
.. 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
|
||||
unknowns are an adapted pair $(Y, Z) \in \mathcal{S}^2 \times \mathcal{H}^2$. The auxiliary
|
||||
process $Z$ is a *non-anticipative hedge*: it makes the equation adapted despite the terminal
|
||||
constraint.
|
||||
|
||||
The primitive `linear_bsde_constant_coeffs` solves the constant-coefficient linear case
|
||||
|
||||
.. math::
|
||||
|
||||
-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 $\Delta t$).
|
||||
|
||||
Mathematical background
|
||||
-----------------------
|
||||
|
||||
**Pardoux–Peng theorem (1990).** If $f$ is uniformly Lipschitz in $(y, z)$ and
|
||||
$\mathbb{E}\!\int_0^T f(s, 0, 0)^2\, ds < \infty$, 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
|
||||
$\Phi : (Y, Z) \mapsto (Y', Z')$ with
|
||||
$Y'_t = \mathbb{E}\bigl[\xi + \int_t^T f(s, Y_s, Z_s)\, ds \bigm| \mathcal{F}_t\bigr]$ and $Z'$
|
||||
obtained by the martingale representation theorem.
|
||||
|
||||
**Closed-form for the linear case.** For $a, b, c$ deterministic the solution is the
|
||||
conditional expectation under a Girsanov-shifted measure:
|
||||
|
||||
.. math::
|
||||
|
||||
Y_t \;=\; \mathbb{E}\!\left[\, \xi\, e^{\int_t^T a(s)\, ds}
|
||||
\;+\; \int_t^T c(s)\, e^{\int_t^s a(r)\, dr}\, ds
|
||||
\,\Big|\, \mathcal{F}_t \right],
|
||||
|
||||
with the Girsanov density $\frac{d\mathbb{Q}}{d\mathbb{P}} = \mathcal{E}\bigl(\int_0^\cdot b(s)\,dW_s\bigr)$.
|
||||
When $b = c = 0$, $a \equiv -\rho$ and $\xi = 1$ this collapses to the analytic ground truth
|
||||
$Y_t = e^{-\rho(T-t)}$ used by the convergence test.
|
||||
|
||||
**Feynman–Kac bridge.** Setting $f(s, y, z) = -r y$ and $\xi = g(X_T)$ for a forward SDE $X$
|
||||
recovers the discounted-payoff PDE: $Y_t = e^{-r(T-t)} \mathbb{E}[g(X_T) \mid \mathcal{F}_t]$.
|
||||
More generally, the markovian BSDE
|
||||
|
||||
.. math::
|
||||
|
||||
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
|
||||
$Y_t = u(t, X_t)$ and $Z_t = \sigma^\top(t, X_t)\nabla u(t, X_t)$.
|
||||
|
||||
**Crank–Nicolson θ-scheme.** On a uniform grid $0 = t_0 < \cdots < t_N = T$ the scheme reads
|
||||
|
||||
.. math::
|
||||
|
||||
Y^N_{t_i} \;=\; \mathbb{E}\!\bigl[\, Y^N_{t_{i+1}} \,\big|\, \mathcal{F}_{t_i}\bigr]
|
||||
\;+\; \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 $Z^N_{t_i} = \Delta t^{-1}\,\mathbb{E}\bigl[Y^N_{t_{i+1}}(W_{t_{i+1}} - W_{t_i})\bigm|\mathcal{F}_{t_i}\bigr]$
|
||||
(discrete Clark–Ocone identity). For $\theta = 1/2$ 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
|
||||
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
|
||||
minimises $\mathbb{E}\bigl[(Y^\theta_T - \xi)^2\bigr]$ over $(Y_0, \theta)$. 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
|
||||
contingent claim $\xi$ and $Z_t$ is the instantaneous hedge ratio. Constraints (transaction
|
||||
costs, portfolio caps, recursive utilities) are absorbed into the driver $f$.
|
||||
* **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
|
||||
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).
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/10_bsde.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/10_bsde.ipynb>`_
|
||||
|
||||
10 — BSDE θ-scheme
|
||||
==================
|
||||
|
||||
@@ -1,9 +1,85 @@
|
||||
Generative calibration — Gaussian MMD loss
|
||||
Generative calibration — Gaussian-MMD loss
|
||||
==========================================
|
||||
|
||||
Maximum-Mean-Discrepancy distance with Gaussian kernel (`mmd_gaussian`). Self-distance is exactly zero; the metric grows monotonically with sample shift.
|
||||
Kernel-based **Maximum Mean Discrepancy** distance (Gretton et al. 2012) — a closed-form,
|
||||
differentiable, distribution-free metric between two empirical samples. Used as the loss
|
||||
function of every generative-calibration loop in `optimiz-rs`.
|
||||
|
||||
.. note:: Companion executed notebook: `17_generative_calibration.ipynb <../../examples/notebooks/17_generative_calibration.ipynb>`_
|
||||
Mathematical background
|
||||
-----------------------
|
||||
|
||||
**Definition.** For a positive-definite kernel $k : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$
|
||||
with reproducing-kernel Hilbert space (RKHS) $\mathcal{H}_k$, the *kernel mean embedding* of a
|
||||
probability measure $P$ is $\mu_P := \mathbb{E}_{X \sim P}[k(X, \cdot)] \in \mathcal{H}_k$.
|
||||
The **squared MMD** is the RKHS distance between embeddings:
|
||||
|
||||
.. math::
|
||||
|
||||
\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*
|
||||
(e.g. Gaussian RBF), $\mathrm{MMD}(P, Q) = 0 \iff P = Q$.
|
||||
|
||||
**U-statistic estimator.** Given i.i.d. samples $\{x_i\}_{i=1}^n$ and $\{y_j\}_{j=1}^m$, the
|
||||
unbiased estimator is
|
||||
|
||||
.. math::
|
||||
|
||||
\widehat{\mathrm{MMD}}^2 \;=\;
|
||||
\frac{1}{n(n-1)}\!\sum_{i \ne i'} k(x_i, x_{i'})
|
||||
\;-\; \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
|
||||
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
|
||||
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
|
||||
$\mathbb{R}$ and $P = \delta$ atomic, the squared MMD is $2 - 2 e^{-\Delta^2 / (2\sigma^2)}$ —
|
||||
smooth, monotone in $|\Delta|$, asymptote $2$ as $\Delta \to \infty$. 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 $Q = \mathcal{N}(\mu_2, \sigma_2^2)$,
|
||||
|
||||
.. math::
|
||||
|
||||
\mathrm{MMD}^2_\sigma(P, Q) \;=\;
|
||||
\frac{\sigma}{\sqrt{\sigma^2 + 2\sigma_1^2}}
|
||||
\;-\; \frac{2\sigma}{\sqrt{\sigma^2 + \sigma_1^2 + \sigma_2^2}}\, e^{-\frac{(\mu_1 - \mu_2)^2}{2(\sigma^2 + \sigma_1^2 + \sigma_2^2)}}
|
||||
\;+\; \frac{\sigma}{\sqrt{\sigma^2 + 2\sigma_2^2}} ,
|
||||
|
||||
giving an exact reference for unit tests.
|
||||
|
||||
**Statistical guarantee.** Gretton et al. (2012, Thm. 12) give the deviation bound
|
||||
$\Pr\!\bigl(\widehat{\mathrm{MMD}}^2 - \mathrm{MMD}^2 > \varepsilon\bigr) \le \exp\bigl(-\varepsilon^2 nm / (8 K^2 (n + m))\bigr)$
|
||||
for $|k| \le K$. Hence MMD detects fixed alternatives at the optimal $n^{-1/2}$ 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 —
|
||||
the reason it is the loss of choice in implicit-generative-model training
|
||||
(generator-loss / score-matching alternatives).
|
||||
|
||||
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
|
||||
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
|
||||
extraction, anomaly detection.
|
||||
* **Model selection.** Replace likelihood ratios when likelihoods are intractable
|
||||
(simulator-based inference, ABC).
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/17_generative_calibration.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/17_generative_calibration.ipynb>`_
|
||||
|
||||
17 — MMD calibration loss
|
||||
=========================
|
||||
|
||||
@@ -1,9 +1,85 @@
|
||||
McKean–Vlasov — propagation of chaos
|
||||
====================================
|
||||
|
||||
Interacting-particle Euler scheme for $dX_t = θ(\bar X_t - X_t) dt + σ dW_t$ (`mean_reverting_mckean_vlasov`). The empirical mean is preserved; the empirical variance approaches the diffusion-only equilibrium.
|
||||
A **McKean–Vlasov SDE** is a stochastic differential equation whose drift and diffusion depend
|
||||
on the *law* of the solution itself:
|
||||
|
||||
.. note:: Companion executed notebook: `14_mckean_vlasov.ipynb <../../examples/notebooks/14_mckean_vlasov.ipynb>`_
|
||||
.. math::
|
||||
|
||||
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
|
||||
|
||||
.. math::
|
||||
|
||||
dX^{i,N}_t \;=\; b\!\Bigl(t, X^{i,N}_t, \tfrac1N\!\sum_{j=1}^N \delta_{X^{j,N}_t}\Bigr)\, dt
|
||||
\;+\; \sigma\!\Bigl(t, X^{i,N}_t, \tfrac1N\!\sum_{j=1}^N \delta_{X^{j,N}_t}\Bigr)\, dW^i_t .
|
||||
|
||||
The primitive shipped here, `mean_reverting_mckean_vlasov`, simulates the canonical example
|
||||
|
||||
.. math::
|
||||
|
||||
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 $X^{i,N}_{k+1} = X^{i,N}_k + \theta(\bar X^N_k - X^{i,N}_k)\Delta t + \sigma\sqrt{\Delta t}\,\xi^i_k$.
|
||||
|
||||
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:
|
||||
|
||||
.. 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 $\rho_t = \mathrm{law}(X_t)$
|
||||
satisfies the *nonlinear* PDE
|
||||
|
||||
.. math::
|
||||
|
||||
\partial_t \rho_t \;+\; \nabla\!\cdot\!\bigl(b(t, x, \rho_t)\, \rho_t\bigr)
|
||||
\;=\; \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
|
||||
SDE, so each marginal is Gaussian with
|
||||
|
||||
.. math::
|
||||
|
||||
\mathbb{E}[X_t] \;=\; \bar X_0,
|
||||
\qquad
|
||||
\mathrm{Var}(X_t) \;=\; \mathrm{Var}(X_0)\, e^{-2\theta t} \;+\; \frac{\sigma^2}{2\theta}\bigl(1 - e^{-2\theta t}\bigr)
|
||||
\;\xrightarrow[t\to\infty]{}\; \frac{\sigma^2}{2\theta}.
|
||||
|
||||
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
|
||||
*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
|
||||
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 $W_2$.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/14_mckean_vlasov.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/14_mckean_vlasov.ipynb>`_
|
||||
|
||||
14 — McKean–Vlasov mean-reverting dynamics
|
||||
==========================================
|
||||
|
||||
@@ -1,9 +1,83 @@
|
||||
PDE — Fokker–Planck, HJB, elliptic Poisson
|
||||
==========================================
|
||||
|
||||
Three CPU-only finite-difference solvers: 1-D forward Fokker–Planck (`fokker_planck_constant`), 2-D explicit HJB (`hjb_quadratic_2d`) and 2-D Poisson SOR (`poisson_2d_zero_boundary`). Each routine is verified against an analytic ground truth.
|
||||
Three CPU-only finite-difference solvers covering the two canonical PDE pillars of stochastic
|
||||
analysis: the **forward** equation for the marginal density of a diffusion (Fokker–Planck),
|
||||
the **backward** equation for an optimally controlled diffusion (Hamilton–Jacobi–Bellman),
|
||||
and a static **elliptic** boundary-value problem (Poisson).
|
||||
|
||||
.. note:: Companion executed notebook: `11_pde.ipynb <../../examples/notebooks/11_pde.ipynb>`_
|
||||
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 $X_t$
|
||||
satisfies the parabolic PDE
|
||||
|
||||
.. math::
|
||||
|
||||
\partial_t \rho \;+\; \partial_x\!\bigl(\mu(t,x)\, \rho\bigr)
|
||||
\;=\; \tfrac12\, \partial^2_{xx}\!\bigl(\sigma^2(t,x)\, \rho\bigr),
|
||||
\qquad \rho(0, \cdot) = \rho_0 .
|
||||
|
||||
For the *pure-diffusion* test ($\mu \equiv 0$, $\sigma^2 \equiv 1$, $\rho_0 = \mathcal{N}(0, 1)$)
|
||||
the analytic Gaussian heat kernel gives $\rho(t, x) = \frac{1}{\sqrt{2\pi(1+t)}}\exp\!\bigl(-\frac{x^2}{2(1+t)}\bigr)$,
|
||||
so the variance grows linearly: $\mathrm{Var}(X_t) = 1 + t$. 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
|
||||
$v(t, x) = \sup_\alpha \mathbb{E}_{t,x}\!\bigl[\int_t^T r(X_s, \alpha_s)\, ds + g(X_T)\bigr]$.
|
||||
Dynamic programming produces
|
||||
|
||||
.. math::
|
||||
|
||||
\partial_t v \;+\; \sup_{a \in \mathcal{A}}\Bigl\{ \mu(x, a) \cdot \nabla v
|
||||
\;+\; \tfrac12\, \mathrm{tr}\!\bigl(\sigma\sigma^\top(x)\, \nabla^2 v\bigr)
|
||||
\;+\; r(x, a) \Bigr\} \;=\; 0,
|
||||
\qquad v(T, x) = g(x).
|
||||
|
||||
`hjb_quadratic_2d` discretises this in 2-D by an explicit finite-difference scheme; the simple
|
||||
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 $\Omega = (0,1)^2$,
|
||||
|
||||
.. 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
|
||||
orthonormal basis with eigenvalues $\lambda_{m,n} = (m^2 + n^2)\pi^2$, so for
|
||||
$f = 2\pi^2 \sin(\pi x)\sin(\pi y)$ the *exact* solution is
|
||||
$u(x, y) = \sin(\pi x)\sin(\pi y)$. `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
|
||||
$\rho \sim 1 - 2\pi h$ — i.e. $O(h^{-1})$ iterations to reach a fixed tolerance, against
|
||||
$O(h^{-2})$ for plain Gauss–Seidel.
|
||||
|
||||
**Probabilistic representation (Feynman–Kac).** Both the parabolic HJB and the elliptic
|
||||
Poisson PDE admit stochastic representations: $u(x) = \mathbb{E}_x\!\bigl[\int_0^{\tau_\Omega} f(X_s)\, ds\bigr]$
|
||||
for the latter, where $\tau_\Omega$ is the first exit time of the diffusion from $\Omega$.
|
||||
This links the PDE solvers above to the BSDE primitives of :doc:`bsde`.
|
||||
|
||||
Why it matters
|
||||
--------------
|
||||
|
||||
* **Density estimation under controlled noise.** Fokker–Planck is the workhorse of
|
||||
non-equilibrium statistical physics, plasma transport, calibration of stochastic-volatility
|
||||
models, and Langevin-based MCMC convergence diagnostics.
|
||||
* **Optimal control & inverse problems.** HJB is the cornerstone of dynamic programming,
|
||||
reinforcement learning (continuous-time policy iteration), and stochastic-control routing.
|
||||
* **Mean-field games.** The MFG fixed point is exactly the coupled system
|
||||
*(backward HJB + forward Fokker–Planck)* with cost depending on the density — building this
|
||||
loop on top of the two solvers above is one of the v2.0 milestones.
|
||||
* **Image processing & PDE-constrained optimisation.** Poisson editing, electric-potential
|
||||
reconstruction, gravitational-potential inversion all reduce to the same elliptic stencil.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/11_pde.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/11_pde.ipynb>`_
|
||||
|
||||
11 — PDE solvers
|
||||
================
|
||||
|
||||
@@ -1,9 +1,83 @@
|
||||
Quadratic-impact control — closed-form Riccati
|
||||
==============================================
|
||||
|
||||
Closed-form Riccati feedback for a controlled 1-D SDE with quadratic running cost (`quadratic_impact_control_py`).
|
||||
Closed-form Riccati feedback for the canonical *single-state, quadratic-cost* linear control
|
||||
problem with running quadratic *impact* penalty.
|
||||
|
||||
.. note:: Companion executed notebook: `13_quadratic_impact.ipynb <../../examples/notebooks/13_quadratic_impact.ipynb>`_
|
||||
Mathematical background
|
||||
-----------------------
|
||||
|
||||
Let $A_t$ be a controlled scalar state driven by an additive control $u_t$ and Gaussian noise.
|
||||
The controller minimises the *finite-horizon quadratic objective*
|
||||
|
||||
.. math::
|
||||
|
||||
J(u) \;=\; \mathbb{E}\!\left[\,\int_0^T \bigl(\,\tfrac{\gamma}{2}\, u_t^2
|
||||
\;+\; \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**
|
||||
and $A_T$ the **terminal penalty** (over-loaded notation: $A_T$ here is the *coefficient*).
|
||||
|
||||
**Hamilton–Jacobi–Bellman.** With value function $v(t, A) = \tfrac12 h(t)\, A^2 + c(t)$, the
|
||||
HJB equation collapses to a scalar Riccati ODE on $h$:
|
||||
|
||||
.. math::
|
||||
|
||||
h'(t) \;=\; \frac{h(t)^2}{\gamma} \;-\; \phi,
|
||||
\qquad
|
||||
h(T) \;=\; A_T .
|
||||
|
||||
The optimal feedback is the linear law
|
||||
|
||||
.. math::
|
||||
|
||||
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.
|
||||
|
||||
**Closed-form solutions.**
|
||||
|
||||
* **Symmetric fixed point** $\gamma = \phi = A_T = 1$: $h(t) \equiv 1$ is the unique solution
|
||||
(RHS vanishes), so the feedback gain is constant $k \equiv 1$. The notebook checks this
|
||||
to machine precision.
|
||||
* **Generic $\phi > 0$.** Writing $\bar h = \sqrt{\gamma \phi}$ for the steady-state and
|
||||
$\rho = \sqrt{\phi / \gamma}$, the Riccati ODE has the closed-form (separation of variables /
|
||||
Bernoulli substitution)
|
||||
|
||||
.. math::
|
||||
|
||||
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 $\bar h = \sqrt{\gamma\phi}$.
|
||||
* **Free of running risk** $\phi = 0$. Then $h'(t) = h(t)^2/\gamma$ 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`.
|
||||
|
||||
**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
|
||||
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).
|
||||
|
||||
Why it matters
|
||||
--------------
|
||||
|
||||
* **Optimal execution.** Almgren–Chriss and its mean-field variants reduce to exactly this
|
||||
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
|
||||
Riccati ODEs; this scalar primitive is the verification kernel against which the matrix
|
||||
solver in :doc:`matrix_riccati` is tested.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/13_quadratic_impact.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/13_quadratic_impact.ipynb>`_
|
||||
|
||||
13 — Quadratic-impact controlled SDE
|
||||
====================================
|
||||
|
||||
@@ -1,9 +1,92 @@
|
||||
Inference — Huber-IRLS drift estimator
|
||||
======================================
|
||||
Inference — Huber-IRLS robust drift estimator
|
||||
=============================================
|
||||
|
||||
Robust drift estimator (`robust_drift`) for $x_{k+1} = x_k + (a + b x_k) Δt + σ ε_k$ via Huber IRLS — resists 5 % heavy-tailed innovations.
|
||||
Heavy-tail-resistant maximum-likelihood estimator for the discrete Ornstein–Uhlenbeck-type model
|
||||
|
||||
.. note:: Companion executed notebook: `16_robust_drift.ipynb <../../examples/notebooks/16_robust_drift.ipynb>`_
|
||||
.. math::
|
||||
|
||||
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
|
||||
plus a fraction $\eta$ 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
|
||||
$y_k = a + b\, x_k + \sigma\, \Delta t^{-1/2}\, \varepsilon_k$. Ordinary least-squares
|
||||
minimises $\sum_k (y_k - a - b x_k)^2$ but its breakdown point is $0$: a single outlier with
|
||||
$|\varepsilon_k| \gg 1$ moves the estimate arbitrarily far.
|
||||
|
||||
**Huber loss & IRLS.** Huber (1964) replaces the quadratic loss by the *piecewise* loss
|
||||
|
||||
.. math::
|
||||
|
||||
\rho_\delta(r) \;=\;
|
||||
\begin{cases}
|
||||
\tfrac12\, r^2, & |r| \le \delta, \\[2pt]
|
||||
\delta\,\bigl(|r| - \tfrac\delta2\bigr), & |r| > \delta,
|
||||
\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
|
||||
as a weighted least-squares problem with weights
|
||||
|
||||
.. math::
|
||||
|
||||
w_k \;=\; \min\!\Bigl(1,\; \frac{\delta}{|r_k|}\Bigr) ,
|
||||
|
||||
so the **Iteratively Reweighted Least-Squares** algorithm reads
|
||||
|
||||
.. math::
|
||||
|
||||
\widehat{(a, b)}^{(t+1)} \;=\; \arg\min_{a, b}\; \sum_k w^{(t)}_k\, (y_k - a - b\, x_k)^2,
|
||||
\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
|
||||
the number of iterations.
|
||||
|
||||
**Choice of the cut-off.** The default $\delta = 1.345 \cdot \hat\sigma$ delivers $95\%$
|
||||
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
|
||||
the ordinary least-squares solution $(\hat a^0, \hat b^0)$ reads
|
||||
|
||||
.. math::
|
||||
|
||||
\binom{\widehat a}{\widehat b}
|
||||
\;=\;
|
||||
\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
|
||||
linearisation shows $\widehat\theta - \theta^\star = O_P(N^{-1/2})$ even in the contaminated
|
||||
model, with asymptotic variance $\sigma^2_\psi / I^2_\psi$ (Huber, *Robust Statistics*, 2004,
|
||||
Thm. 7.7).
|
||||
|
||||
**Connection with Malliavin calculus.** The driver $a + b\, x$ 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.
|
||||
|
||||
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
|
||||
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.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/16_robust_drift.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/16_robust_drift.ipynb>`_
|
||||
|
||||
16 — Robust drift estimation
|
||||
============================
|
||||
|
||||
@@ -1,9 +1,84 @@
|
||||
Stochastic control — switching, Pontryagin, two-sided intensities
|
||||
=================================================================
|
||||
|
||||
Three primitives: discrete-time optimal switching (`optimal_switching_dp`), 1-D Pontryagin LQR shooting (`pontryagin_lqr`) and the bilateral intensity controller (`two_sided_intensities`).
|
||||
Three complementary primitives covering the discrete and continuous worlds of stochastic
|
||||
control: dynamic-programming **optimal switching** (Snell envelope), the continuous-time
|
||||
**Pontryagin–Bismut maximum principle** for the linear-quadratic regulator, and a **two-sided
|
||||
intensity controller** for jump processes.
|
||||
|
||||
.. note:: Companion executed notebook: `12_stochastic_control.ipynb <../../examples/notebooks/12_stochastic_control.ipynb>`_
|
||||
Mathematical background
|
||||
-----------------------
|
||||
|
||||
**1. Optimal switching as a Snell envelope.** Let $(Y^i_k)_{k, i}$ be the running rewards in
|
||||
mode $i \in \{1, \dots, M\}$ and $c_{ij}$ the cost of switching from $i$ to $j$. The value
|
||||
function $V_k(i)$ satisfies the backward dynamic-programming recursion
|
||||
|
||||
.. math::
|
||||
|
||||
V_N(i) = g(i),
|
||||
\qquad
|
||||
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
|
||||
$i \neq 1$ and $V_k(1) = N - k + 1$ — 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
|
||||
$J(u) = \mathbb{E}\!\bigl[\int_0^T (q X_t^2 + r u_t^2)\, dt + s_T X_T^2\bigr]$, the
|
||||
adjoint variable $P_t$ solves the **matrix Riccati ODE**
|
||||
|
||||
.. 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
|
||||
$a = q = 0$, $b = r = s_T = 1$, $T = 1$ the ODE simplifies to
|
||||
$\dot P_t = P_t^2$, whose closed-form solution is
|
||||
|
||||
.. math::
|
||||
|
||||
P_t \;=\; \frac{1}{1 + (T - t)} ,
|
||||
\qquad
|
||||
P(0) = \tfrac12 .
|
||||
|
||||
The primitive `pontryagin_lqr` reproduces this with relative error below $10^{-3}$ for
|
||||
$N = 2000$ steps (the symmetric Strang splitting is second-order in $\Delta t$).
|
||||
|
||||
**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*
|
||||
:math:`\delta_\pm(\lambda) = \alpha_\pm + \kappa_\pm \lambda` and value-function jumps
|
||||
:math:`\Delta V_\pm`, the instantaneous Hamiltonian is
|
||||
|
||||
.. math::
|
||||
|
||||
\sup_{\lambda_\pm \ge 0}\!\Bigl[\,\lambda_+\bigl(\delta_+(\lambda_+) - \Delta V_+\bigr)
|
||||
\;+\; \lambda_-\bigl(\delta_-(\lambda_-) - \Delta V_-\bigr)\Bigr],
|
||||
|
||||
and the first-order condition gives the closed-form maximiser
|
||||
|
||||
.. math::
|
||||
|
||||
\lambda^*_\pm \;=\; \max\!\Bigl(0,\; \frac{\alpha_\pm - \Delta V_\pm}{2\, \kappa_\pm}\Bigr).
|
||||
|
||||
The quantity :math:`\Delta V_\pm` is the (estimated) marginal value of an additional event;
|
||||
`two_sided_intensities` returns :math:`(\lambda^*_+, \lambda^*_-)` in closed form, which is what
|
||||
lets the broader optimal-execution loop run in real time.
|
||||
|
||||
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.
|
||||
* **Two-sided intensity control** is the closed-form heart of optimal market making
|
||||
(Avellaneda–Stoikov 2008, Cartea–Jaimungal–Penalva 2015) and limit-order placement.
|
||||
|
||||
.. note::
|
||||
📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/12_stochastic_control.ipynb>`_
|
||||
· `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/12_stochastic_control.ipynb>`_
|
||||
|
||||
12 — Stochastic control
|
||||
=======================
|
||||
@@ -102,7 +177,7 @@ Closed-form Riccati for $a=q=0$, $b=r=s_T=1$, $T=1$ is $P(t) = 1/(1 + (T - t))$,
|
||||
Two-sided intensity control
|
||||
---------------------------
|
||||
|
||||
Affine premium $δ_±(λ) = α_± + κ_± λ$. First-order condition: $\lambda^*_\pm = \max(0, (α_\pm - ΔV_\pm) / (2 κ_\pm))$.
|
||||
Affine premium :math:`\delta_\pm(\lambda) = \alpha_\pm + \kappa_\pm \lambda`. First-order condition: :math:`\lambda^*_\pm = \max(0, (\alpha_\pm - \Delta V_\pm) / (2 \kappa_\pm))`.
|
||||
|
||||
.. code-block:: python
|
||||
|
||||
|
||||
Reference in New Issue
Block a user