Solving non-Markovian stochastic control problems driven by Wiener functionals

Abstract

In this article, we present a general methodology for stochastic control problems driven by the Brownian motion filtration including non-Markovian and nonsemimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a numerical scheme for computing near-optimal controls associated with controlled Wiener functionals via a finite-dimensional approximation procedure. The approach does not require functional differentiability assumptions on the value process and ellipticity conditions on the diffusion components. The general convergence of the method is established under rather weak conditions for distinct types of non-Markovian and nonsemimartingale states. Explicit rates of convergence are provided in case the control acts only on the drift component of the controlled system. Near-closed/open-loop optimal controls are fully characterized by a dynamic programming algorithm and they are classified according to the strength of the possibly underlying non-Markovian memory. The theory is applied to stochastic control problems based on path-dependent SDEs and rough stochastic volatility models, where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for nonlinear path-dependent SDEs driven by fractional Brownian motion with exponent $\mathit{H}\in (0,\frac{1}{2})$ is also discussed. Finally, we present a simple numerical example to illustrate the method.