Fuzzy stochastic differential equations, fuzzy stochastic processes, Bihari's inequality, existence and uniqueness, averaging method, 65C30, Mathematics, QA1-939, Engineering (General). Civil engineering (General), TA1-2040

The fuzzy stochastic differential equations with Markovian switching are considered. First, under the [Formula: see text] assumptions, the existence and uniqueness theorem for the aforementioned equations is given by means of stopping time techniques, discretization method, and the Gronwall-Bellman inequality. Then the boundedness of the second moment of the solutions for such equations is established. Furthermore, by using the averaging method and stochastic analysis, as well as the Bihari's inequality, this paper mainly demonstrates the asymptotic relationship between the solutions of the original equations and the corresponding averaged equations in terms of mean square and probability. Finally, the theoretical result of the averaging method is illustrated by an example.