Computation of nash equilibrium pairs of a stochastic differential game

Abstract

Consider the random motion of two points Me and Mp in an open and bounded domain D0 in the plane. Each of the velocities, u = (u1 u2)T of Me and v = (v1, v2)T of Mp, are perturbed by a corresponding R2-valued Gaussian white noise. Let A and Dc be two disjoint closed subsets of D0. Suppose that at t = 0, Me is in A and Mp is anywhere in D0. Denote by ℰ1 and ℰ2 the following events: ℰ1 = {Mp intercepts Me in A before Me reaches the set Dc and before either Me or Mp has left D0}, and ℰ2 = {Me reaches the set Dc before being intercepted by Mp, while Mp is in A, and before either Mp or Me has left D0}. The problem dealt with here is to find a pair of velocity strategies (u*, v*) such that, in the sense of a Nash equilibrium point, the probabilities Prob(ℰ1) and Prob(ℰ2) will both be maximized on a given class of velocity strategies (u, v). Sufficient conditions on (u*, v*) are derived which require the existence of a smooth solution (V,Q) to a pair of coupled non-linear partial differential equations. A finite-difference scheme for solving these equations is suggested, and two examples are treated in detail.