Viability property for multi-dimensional stochastic differential equation and its applications to comparison theorem

Abstract

In this paper, we focus on the problem of viability property for the multi-dimensional stochastic differential equations, where the coefficients of the stochastic differential equations may be random and discontinuous in the time variable. First, we establish a necessary and sufficient condition for viability property with respect to a given non-empty closed convex set. And furthermore, with the help of this result, we also study the multi-dimensional comparison theorems about stochastic differential equations.