Explicit strategies for some linear and nonlinear stochastic differential games

Abstract

Some two person stochastic differential games are formulated and solved that are described by either linear or nonlinear stochastic equations. For suitable payoff functionals, explicit optimal control strategies that are Nash equilibria are obtained for the two players in these games. The solution approach does not require solving the Isaacs equations or backward stochastic differential equations.The linear stochastic equations can have stochastic coefficients and state dependent noise. The non-linear stochastic equations evolve in the unit n-sphere for arbitrary dimension n. An interpretation of the determination of the strategies is given in terms of the decomposition of submartingales and supermartingales as sums or differences of martingales and increasing processes.