Stochastic Differential Games in a Non-Markovian Setting

TL;DR

This paper explores stochastic differential games driven by fractional Brownian motion, providing explicit Nash equilibria solutions in a non-Markovian setting where traditional HJB methods are not applicable.

Contribution

It introduces a novel approach using fractional noise calculus to find Nash equilibria in non-Markovian stochastic differential games, extending the analysis beyond Markovian models.

Findings

Explicit Nash equilibria derived for fractional Brownian motion modulated games

Applicable to financial models involving institutional investors and stock price dynamics

Framework adaptable to general Gaussian stochastic differential games

Abstract

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.