Differential Games for Markov Jump Linear Systems with Fractional Brownian Perturbation

Abstract

The optimal control problem of Markov Jump linear systems (MJLS) driven by fractional Brownian motion (fBm) depends upon a single objective function (or a single decision maker). However, there are situations when there may be more than one decision maker, each having one's own payoff function to be optimized, subject to a set of differential equations. Therefore, using a natural generalization of (one player) stochastic control problem, stochastic differential games are considered here in a non-Markovian setting. We study two stochastic differential games of continuous-time MJLS in finite-time horizon in which the modulating process is a fBm with Hurst parameter $H \in ( 1 / 2 ,1)$ and the state space of the Markov chain is finite. First, a two person non-zero-sum stochastic differential game with a quadratic cost functional for each player is discussed. In the sequel, a non-cooperative, two person, zero-sum, stochastic differential game with a quadratic payoff for the two players is formulated and solved. In both cases, the optimal strategies are given explicitly.