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.