A solvable stochastic differential game in the two-sphere

In this paper a two person zero sum stochastic differential game is formulated and explicitly solved where the state of the game evolves in a two dimensional sphere. The game is described by a stochastic equation that is the sum of the control strategies of the two players and a Brownian motion in the two-sphere. The problem formulation uses the property that the two-sphere is a rank one compact symmetric space. For a suitable payoff that reflects the geometry of the compact symmetric space, a direct method provides optimal control strategies. This approach does not require solving either the Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations. The value of the game is also given. The game problems include both finite and infinite time horizons. Some extensions of this model to other solvable stochastic differential games is noted.

I. Introduction

Two person, zero sum stochastic differential games developed as a natural generalization of (one player) stochastic control problems and minimax control problems. Isaacs [9] obtained nonlinear partial differential equations that determine the lower and the upper values of the game. If these two values of the game are equal then the game is said to have a value and the two nonlinear partial differential equations become one. It is difficult to solve the Hamilton-Jacobi-Isaacs equation, though some special cases are known such as a linear-quadratic stochastic differential game. It is important to determine a family of explicitly solvable stochastic differential games to provide insight for other stochastic differential games and to provide test cases for numerical algorithms as well as the intrinsic importance of solving stochastic differential games.