Gaussian path approximation

Gaussian path approximation is the primary variational method used in GSBM to solve the CondSOC problem for general nonlinear state costs $V_t$. The idea is to approximate the optimal conditional probability path $p_t(X_t|x_0,x_1)$ as a Gaussian pinned at the two endpoints:

$p_t(X_t|x_0,x_1)\approx \mathcal{N}({\mu }_t,{\gamma }_t^2{I}_d),{\mu }_0=x_0,{\mu }_1=x_1,{\gamma }_0={\gamma }_1=0$

The time-varying mean ${\mu }_t\in {\mathbb{R}}^d$ and standard deviation ${\gamma }_t\in \mathbb{R}$ are parameterized as splines with $K\le 30$ control points, ensuring the boundary conditions are satisfied for any control point values. This yields an analytic conditional drift $u_t(X_t|x_0,x_1)={\partial }_t{\mu }_t+a_t(X_t-{\mu }_t)$ where $a_t=\frac{1}{{\gamma }_t}({\partial }_t{\gamma }_t-\frac{{\sigma }^2}{2{\gamma }_t})$ is a time-varying scalar. Despite potentially complex paths ${\mu }_t$ and ${\gamma }_t$, the underlying SDE remains linear, and the drift is a gradient field — hence kinetically optimal among all drifts generating the same conditional density.

The approach is directly inspired by the analytic solution for quadratic $V_t$ (Lemma 3 of GSBM), where the optimal path is exactly Gaussian with coefficients involving $\sinh$ and $\cosh$ of $\eta =\sigma \sqrt{2\alpha }$. The Brownian bridge is recovered as $\alpha \to 0$ (linear interpolation with ${\gamma }_t=\sigma \sqrt{t(1-t)}$), and the straight line (rectified flow) as further $\sigma \to 0$. This hierarchy connects to the stochastic interpolant framework of Albergo et al. (2023), where the general spatially linear interpolation $x_t=\alpha (t)x_0+\beta (t)x_1+\gamma (t)z$ encompasses the same design space.

Key Details

• Spline parameterization: ${\mu }_t=\text{Spline}(t;x_0,\{X_{t_k}\},x_1)$, ${\gamma }_t=\text{Spline}(t;0,\{{\gamma }_{t_k}\},0)$
• $K\le 30$ control points — drastically fewer parameters than caching full SDE trajectories
• Simulation-free: optimization uses independent samples from $p_{t|0,1}$ (closed-form Gaussian)
• Initialization: $\{X_{t_k}\}$ from current drift $u_t^{\theta }$, ${\gamma }_{t_k}=\sigma \sqrt{t_k(1-t_k)}$ (Brownian bridge)
• Parallelizable over all $(x_0,x_1)$ pairs in a batch
• Can be further debiased via path integral resampling
• Efficient covariance computation via 1D ODE for Alg. 6

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method