Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3

Tohoku Math. J.
67 (2015), 349-381

Title SET-VALUED AND FUZZY STOCHASTIC DIFFERENTIAL EQUATIONS IN M-TYPE 2 BANACH SPACES

Author Marek T. Malinowski

(Received May 31, 2013, revised May 7, 2014)
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.

Mathematics Subject Classification. Primary 60H20; Secondary 60H05, 28B20, 45R05, 93E03.
Key words and phrases. Set-valued stochastic integral, stochastic integration in Banach spaces, set-valued stochastic differential equation, set-valued stochastic integral equation, existence and uniqueness of solution, fuzzy stochastic integral, fuzzy stochastic differential equation.

To the top of this page

Back to the Contents