Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces

Abstract

In this paper we study set-valued stochastic differential equations in M-type 2 Banach spaces. Their drift terms and diffusion terms are assumed to be set-valued and single-valued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define set-valued stochastic Lebesgue integral in a Banach space. This integral is a set-valued random variable. We state its properties such as additivity with respect to the interval of integration, continuity as a function of the upper limit of integration, integrable boundedness. The existence and uniqueness of solution to set-valued differential equations in M-type 2 Banach space is obtained by a method of successive approximations. We show that the approximations are uniformly bounded and converge to the unique solution. A distance between $n$th approximation and exact solution is estimated and a continuous dependence of solution with respect to the data of the equation is proved. Finally, we construct a fuzzy stochastic Lebesgue integral in a Banach space and examine fuzzy stochastic differential equations in M-type 2 Banach spaces. We investigate properties like those in set-valued cases. All the results are achieved without assumption on separability of underlying sigma-algebra.