Fuzzy and Set-Valued Stochastic Differential Equations With Local Lipschitz Condition

Abstract

We are concerned with the fuzzy stochastic differential equations driven by multidimensional Brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises. Under the uniform Lipschitz condition, we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations. Next we show, assuming the Lipschitz condition is satisfied only locally, that these equations have a unique solution. The fact that the solution is bounded is also proved. We conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set-valued stochastic differential equations with local Lipschitz condition.