k=1 Gk(ω) · 1(uk,vk](s), ω ∈ Ω, s ≥ 0, where n ∈ N, 0 ≤ uk < vk, and Gk ∈ L(Fuk). We let Le denote the class of all such integrands. Notice that each H ∈ Le satisfies (i) s → Hs(ω) is a left-continuous step function for each ω ∈ Ω, (ii) Hs ∈ L(Fs) for each s ≥ 0, (iii) viewed as a mapping from Ω × [0, t] to R, (ω, s) → Hs(ω) is Ft ⊗B[0,t] measurable for each t > 0, and (iv) E[ ∫ t 0 H s ds] < ∞...

In this paper, we introduce an efficient method based on Haar wavelet to approximate a solutionfor the two-dimensional linear stochastic Fredholm integral equation. We also give an example to demonstrate the accuracy of the method.  

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. particular, prove $L^p$-extrapolation results under a H\"ormander condition on the Sparse domination and sharp weighted bounds are obtained Dini kernel, leading to version of solution $A_2$-conjecture. The applied obtain $p$-independence maximal $L^p$-regularity both in complex re...

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 pr...

A stochastic differential equation model is considered for nonlinear oscillators under excitations of combined Gaussian and Poisson white noise. Since the solutions of stochastic differential equations can be interpreted in terms of several types of stochastic integrals, it is sometimes confusing about which integral is actually appropriate. In order for the energy conservation law to hold unde...

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued ...

In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique give...