A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations
This provides a theoretical framework for optimal control of path-dependent SDEs, addressing a known bottleneck in non-Markovian stochastic control.
The paper develops a quasi-sure approach for controlling non-Markovian SDEs by constructing a value process under a family of singular measures, characterized by a second-order backward SDE, generalizing G-expectation to SDEs.
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.