conditional stochastic optimal control

Conditional Stochastic Optimal Control (CondSOC) is the key computational primitive of the GSBM algorithm. It arises from the decomposition of the generalized Schrödinger bridge problem: when the boundary coupling $p_{0,1}$ is fixed, the GSB objective factorizes (by Fubini’s theorem) into a mixture of independent SOC problems, one for each pair $(x_0,x_1)\sim p_{0,1}$:

${\min }_{u_{t|0,1}}J:={\int }_0^1{\mathbb{E}}_{p_t(X_t|x_0,x_1)}\left[\frac{1}{2}\Vert u_t(X_t|x_0,x_1){\Vert }^2+V_t(X_t)\right]dt$ subject to $d{X}_t=u_t(X_t|x_0,x_1)dt+\sigma d{W}_t$, $X_0=x_0$, $X_1=x_1$.

This differs from the full GSB only in replacing distributional boundary conditions $(\mu ,\nu )$ with fixed endpoints $(x_0,x_1)$, making it much more tractable. The optimal control satisfies $u_t^{\ast }={\sigma }^2\nabla \log {\Psi }_t$ where ${\Psi }_t$ solves a Hamilton-Jacobi-Bellman PDE with a killing term from $V_t$.

For special cases: when $V_t=0$, the CondSOC solution is the Brownian bridge; when $V_t$ is quadratic ($V(x)=\alpha \Vert \sigma x{\Vert }^2$), the solution is a Gaussian path with hyperbolic coefficients (Lemma 3 of GSBM). For general nonlinear $V_t$, GSBM employs Gaussian path approximation with spline-parameterized mean and variance, optionally refined by path integral control.

Key Details

• Factorizes GSB into per-pair SOC problems via law of total expectation + Fubini
• Endpoints are fixed points, not distributions — much simpler than full GSB
• $V_t=0$ solution = Brownian bridge (recovers DSBM)
• $V_t$ quadratic: analytic Gaussian path with $c_t,e_t,{\gamma }_t$ involving $\sinh /\cosh$ of $\eta =\sigma \sqrt{2\alpha }$
• General $V_t$: Gaussian path approximation (Alg. 3) + optional path integral resampling (Alg. 4)
• Parallelizable over batches of $(x_0,x_1)$ pairs
• Simulation-free: only independent samples from $p_{t|0,1}$ needed

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