Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method

TL;DR

This paper develops a stochastic Pontryagin maximum principle for systems driven by Gaussian rough paths, enabling a new indirect shooting method that significantly improves convergence speed in stochastic control problems.

Contribution

It introduces a novel PMP for rough differential equations driven by Gaussian paths and proposes the first indirect shooting method for nonlinear stochastic control.

Findings

The PMP applies to systems driven by Brownian motion without relying on forward-backward SDEs.

The error bounds for RDE solutions are derived using recent Gaussian rough path results.

The proposed shooting method converges 10 times faster than direct methods in a stabilization task.

Abstract

We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle (PMP) applies to systems following stochastic differential equations (SDE) driven by Brownian motion, yet it does not rely on forward-backward SDEs and involves the same Hamiltonian as the deterministic PMP. The proof consists of first deriving various integrable error bounds for solutions to nonlinear and linear RDEs by leveraging recent results on Gaussian rough paths. The PMP then follows using standard techniques based on needle-like variations. As an application, we propose the first indirect shooting method for nonlinear stochastic optimal control and show that it converges 10x faster than a direct method on a stabilization task.