A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations

TL;DR

This paper develops a quasi-sure framework for controlling non-Markovian stochastic differential equations with path-dependent coefficients, using second order backward SDEs to characterize the value process and generalize G-expectation.

Contribution

It introduces a novel quasi-sure approach to control non-Markovian SDEs, linking the value process to second order backward SDEs and extending G-expectation.

Findings

Characterization of the value process via second order backward SDEs

Generalization of G-expectation to controlled SDEs

Framework for path-dependent stochastic control

Abstract

We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This value process is characterized by a second order backward SDE, which can be seen as a non-Markovian analogue of the Hamilton-Jacobi-Bellman partial differential equation. Moreover, our value process yields a generalization of the G-expectation to the context of SDEs.