Stochastic Near-Optimal Controls for Path-Dependent Systems

TL;DR

This paper introduces a pathwise methodology for approximating near-optimal controls in complex stochastic systems driven by non-Markovian processes, applicable to financial models and path-dependent SDEs.

Contribution

It develops a general, non-differentiability reliant framework for control problems involving non-Markovian and non-semimartingale processes, expanding the scope of stochastic control theory.

Findings

Pathwise method effectively characterizes near-optimal controls.

Applicable to path-dependent SDEs with degenerated diffusion.

Constructs near-optimal controls in non-Markovian portfolio optimization.

Abstract

In this article, we present a general methodology for control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete pathwise method for characterizing and computing near-optimal controls for abstract controlled Wiener functionals. The theory does not require ad hoc functional differentiability assumptions on the value process and elipticity conditions on the diffusion components. The analysis is pathwise over suitable finite dimensional spaces and it is based on the weak differential structure introduced by Le\~ao, Ohashi and Simas jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion is also discussed. We finally provide an application in the context of financial mathematics. Namely, we construct near-optimal controls in a non-Markovian portfolio optimization problem.