Stochastic equations with discontinuous drift

Abstract

We study stochastic differential equations, dx = adt + adfl where ft denotes a Brownian motion. By relaxing the definition of solutions we are able to prove existence theorems assuming only that a is measurable, a is continuous and that both grow linearly at infinity. Nondegeneracy is not assumed. The relaxed definition of solution is an extension of A. F. Filippov's definition in the deterministic case. When a is constant we prove one-sided uniqueness and approximation theorems under the assumption that a satisfies a one-sided Lipschitz condition. We consider the stochastic equation dx = a dt +Jg d/. If the drift coefficient, a, and diffusion matrix, g, are continuous on Rd+1, then solutions are known to exist [1]. If a is assumed to be nondegenerate, i.e. u&g* is positive definite, then existence of a solution can be proved assuming that a is bounded and measurable [9]. Moreover, consideration of the one-dimensional example a =0, a(t, x) = sgn (x) shows that continuity is necessary if degeneracy is allowed. However, in the deterministic case (ug=0), A. F. Filippov [2] has shown that a natural and fruitful theory for discontinuous direction fields is possible if the definition of solution is relaxed. In this paper we bring Filippov's ideas into the stochastic context. This results in a fairly general existence theorem (Theorem 3). We were less successful in our treatment of uniqueness, however. That result (Theorem 4) is limited to the case of constant diffusion matrix. Our last result (Theorem 5) concerns approximation of relaxed solutions by solutions of nondegenerate equations. It is a pleasure to acknowledge J. Goldstein with whom I have had several fruitful conversations. I also thank the referee for bringing the work of Prohorov to my attention. 1. Existence of relaxed solutions. We shall denote points of Rd by x= (xl,. . ., xd) and points of Rd+l by (t, x)=(t, x1, . . ., xd). =x1yl + +XdYa denotes the inner product in Rd and xl =VK . Received by the editors June 8, 1970. AMS 1970 subject classifications. Primary 34F05, 60H10, 60H20.