Mixed set-valued stochastic differential equations: Existence, uniqueness and averaging principle

Abstract

The objective of this paper is to investigate mixed set-valued stochastic differential equations with fractional Brownian motion, where the diffusion term is also set-valued. Under the non-Lipschitz continuity conditions, firstly, some new and reliable lemmas about the set-valued stochastic integral are provided. Secondly, we justify the existence and uniqueness of solutions to considered equations by successive approximations, stochastic analysis, and fractional calculus. Moreover, the solution converges to that of the averaged equations in the sense of mean square and probability. Finally, two examples are provided to verify our theoretical results.