Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method

2025-02-10

AI Summary

This paper investigates stochastic optimal control problems driven by Gaussian rough paths, aiming to establish a Pontryagin maximum principle (PMP) independent of forward–backward stochastic differential equations (SDEs). Methodologically, it pioneers the integration of rough path theory with stochastic PMP, constructing a stochastic optimal control framework compatible with classical deterministic Hamiltonian structures; introduces a rigorous derivation pathway based on spike variations and linear/nonlinear rough differential equation (RDE) error estimates; and designs the first indirect shooting method tailored for rough systems. Theoretically, it establishes the first SDE-free stochastic PMP. Numerical experiments demonstrate that the proposed method achieves a tenfold faster convergence rate than direct methods in stabilization tasks, significantly enhancing both computational efficiency and theoretical applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Stochastic optimal control
• Rough differential equations
• Indirect shooting method

Innovation

• Rough differential equations optimization
• Indirect shooting method
• Gaussian rough paths application