Dual and backward SDE representation for optimal control of non-Markovian SDEs

TL;DR

This paper introduces a novel dual and backward SDE framework for solving non-Markovian stochastic control problems with path-dependent coefficients without ellipticity assumptions, extending classical methods to a broader class of stochastic systems.

Contribution

It develops a controls randomization approach and characterizes the value function via a backward SDE with nonpositive jumps, generalizing the Hamilton-Jacobi-Bellman equation to non-Markovian settings.

Findings

Reformulation of the value function under dominated measures.

Representation of the value function via a backward SDE with nonpositive jumps.

Extension of the framework to G-expectation for path-dependent control problems.

Abstract

We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton-Jacobi-Bellman equation, and an extension to $G$ expectation.