Existence theorems for viable solutions of stochastic differential equations

Abstract

We study the existence of solutions of the stochastic differential equation dx(t) = f (t, x(t))dt + g(t, x(t))dW (t) (1) satisfying the condition x(t) ∈ K(t, x(t)), t ∈ [0, T ]. (2) An existence theorem for such solutions, which are said to be viable [1, 2], for the case in which the functions f and g satisfy the Lipschitz condition with respect to t and x and some stochastic tangential condition holds was proved in [1]. A similar theorem was proved in [2] for stochastic differential inclusions under conditions permitting one to use the Ky Fan fixed point theorem. Unlike [1], we consider system (1), (2) with Borel measurable functions f and g and with a mapping K depending on the state variables and use a stochastic tangential condition that differs from the similar condition in [1]. In the present paper, we prove existence theorems for weak and strong solutions of system (1), (2); moreover, solutions of Eq. (1) are understood as solutions of some stochastic inclusion corresponding to this equation.