Stochastic Differential Games in a Non-Markovian Setting

Stochastic di erential games are considered in a non-Markovian setting. Typically, in stochastic di erential games the modulating process of the di usion equation describing the state ow 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 nd the Nash equilibria explicitly. Although fractional Brownian motion is taken as the modulating process because of its versatility in modeling in the elds of nance and networks, the approach in this paper has the merit of being applicable to more general Gaussian stochastic di erential games with only slight conceptual modi cations. This work has applications in nance to stock price modeling which incorporates the e ect of institutional investors, and to stochastic di erential portfolio games in markets in which the stock prices follow di usions modulated with fractional Brownian motion. AMS subject classi cations. 91A15, 91A23, 60G15, 60G18, 60H40