Maximum Principle of Discrete Stochastic Control System Driven by Both Fractional Noise and White Noise

Yuecai Han,Zheng Li

doi:10.1155/2020/1959050

Abstract

In this paper, we investigate the necessary optimality conditions of the discrete stochastic optimal control problems driven by both fractional noise and white noise. Here, the admissible control region is not necessarily convex. The corresponding variational inequalities are obtained by applying the classical variation method and Malliavin calculus. We also apply the stochastic maximum principle to a linear-quadratic optimal control problem to illustrate the main result.

Highlights

IntroductionWith the development of the optimal control theory, some researchers began to work on the discrete case by following the Pontryagin maximum principle for continuous optimal control problems
We consider a stochastic control problem for state process driven by both general white noise and fractional noise withHurst parameter H ∈ ((1/2), 1)
With the development of the optimal control theory, some researchers began to work on the discrete case by following the Pontryagin maximum principle for continuous optimal control problems

Summary

Introduction

With the development of the optimal control theory, some researchers began to work on the discrete case by following the Pontryagin maximum principle for continuous optimal control problems. With the development of the fractional calculus, Han et al [10] obtained a maximum principle for the stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H > (1/2)), and the maximum principle involves Malliavin derivatives. Lin and Zhang [12] developed a maximum principle for optimal control of discrete-time stochastic systems, and the admissible control region was nonconvex.

Preliminaries

The Maximum Principle

Applications to the LinearQuadratic Problem

Conclusion