Existence theorems for solutions of stochastic differential equations with discontinuous right-hand sides

Abstract

We prove existence theorems for weak and strong solutions of the stochastic differential equation dx(t) = f(t,x(t))dt + g(t,x(t))dW(t) (1) with a discontinuous function f : R+ • R d --~ R d and a continuous bounded function g : R+ • R d --~R d• By [1, 2], weak solutions of system (1) exist if the mappings f and g are bounded, measurable, and the closure of the intersection of the degeneracy set {(t, x) I fu(t,x)( det a(r, y ) ) - l d r dy = oc for each open neighborhood U(t, x) of the point (t, x) } of the mapping g with the set of discontinuity points of f or g lies in the set of zeros f and g [here a(t, x) = g(t, x)gT(t, X), and T stands for transposition]