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Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3
Tohoku Math. J.
67 (2015), 349381

Title
SETVALUED AND FUZZY STOCHASTIC DIFFERENTIAL EQUATIONS IN MTYPE 2 BANACH SPACES
Author
Marek T. Malinowski
(Received May 31, 2013, revised May 7, 2014) 
Abstract.
In this paper we study setvalued stochastic differential equations in Mtype 2 Banach spaces. Their drift terms and diffusion terms are assumed to be setvalued and singlevalued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define setvalued stochastic Lebesgue integral in a Banach space. This integral is a setvalued 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 setvalued differential equations in Mtype 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 Mtype 2 Banach spaces. We investigate properties like those in setvalued cases. All the results are achieved without assumption on separability of underlying sigmaalgebra.
Mathematics Subject Classification.
Primary 60H20; Secondary 60H05, 28B20, 45R05, 93E03.
Key words and phrases.
Setvalued stochastic integral, stochastic integration in Banach spaces, setvalued stochastic differential equation, setvalued stochastic integral equation, existence and uniqueness of solution, fuzzy stochastic integral, fuzzy stochastic differential equation.


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