ar X iv : c s / 05 01 05 2 v 1 [ cs . I T ] 2 1 Ja n 20 05 STOCHASTIC DIFFERENTIAL GAMES IN A NON - MARKOVIAN SETTING

Stochastic differential games are considered in a non-Markovian setting. Typically, in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be Markovian. Then Nash equilibria or other types of solution such as Pareto equilibria are constructed using Hamilton-Jacobi-Bellman (HJB) equations. But in a non-Markovian setting the HJB method is not applicable. To examine the non-Markovian case, this paper considers the situation in which the modulating process is a fractional Brownian motion. Fractional noise calculus is used for such models to find the Nash equilibria explicitly. Although fractional Brownian motion is taken as the modulating process because of its versatility in modeling in the fields of finance and networks, the approach in this paper has the merit of being applicable to more general Gaussian stochastic differential games with only slight conceptual modifications. This work has applications in finance to stock price modeling which incorporates the effect of institutional investors, and to stochastic differential portfolio games in markets in which the stock prices follow diffusions modulated with fractional Brownian motion. 1. Introduction. The study of stochastic differential games with controls is a part of game theory that is relatively unknown, even though it has significant potential for application as noted by Øksendal and Reikvam [29]. Prior work in this area has focused on the examination of such games in a Markovian setting (see below). In this paper we will study a type of non-Markovian stochastic differential game. In particular, we will consider a game in which the one-dimensional state X t follows the following stochastic differential equation: