Stochastic differential inclusions of semimonotone type in Hilbert spaces

In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic processes with respect to the cylindrical Brownian motion on separable Hilbert spaces. Then, we generalize the existence results for differential inclusions in [H. Abedi and R. Jahanipur, Nonlinear differential inclusions of semimonotone and condensing type in Hilbert spaces, textit{Bull. Korean Math. Soc.}, {52} (2015), no. 2, 421--438.] to the corresponding stochastic differential inclusions using the methods discussed in [R. Jahanipur, Nonlinear functional differential equations of monotone-type in Hilbert spaces, {it Nonlinear Analysis} {bf 72} (2010), no. 3-4, 1393--1408, R. Jahanipur, Stability of stochastic delay evolution equations with monotone nonlinearity, {it Stoch. Anal. Appl.}, {bf 21} (2003), 161--181, and R. Jahanipur, Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions, {it J. Differential Equations} {bf 248} (2010), no. 5, 1230--1255.]