Peng's Maximum Principle for a Stochastic Control Problem Driven by a Fractional and a Standard Brownian Motion

TL;DR

This paper develops a maximum principle for stochastic control systems driven by both fractional and standard Brownian motions, extending classical results to systems with fractional noise.

Contribution

It introduces a novel maximum principle for control problems involving fractional Brownian motion with Hurst parameter less than 1/2, using an anticipative Girsanov transformation.

Findings

Derived a maximum principle for the combined fractional and standard Brownian motion system.

Established a stochastic variational inequality as a generalization of classical results.

Extended stochastic control theory to systems with fractional noise.

Abstract

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.