Lebesgue-Stieltjes Integrals Of Fuzzy Stochastic Processes

This article mainly studies Lebesgue-Stieltjes integrals of fuzzy stochastic processes with respect to finite variation processes.At first, let (Ω,T,P) be a complete probability space,{.F,t∈[0,T]} be a σ-field satisfying usual condition, G={Gt,t ∈[0,T]} an Ft-adapted fuzzy stochastic process and A={At,t∈[0,T]}an Ft-adapted real valued finite variation process. We will define the Lebesgue-Stieltjes integral of fuzzy stochastic process G with respect to finite variation processes A denoted by ∫0 t Gs (ω)dAs (ω) for each t>0 by using the selection method, which is direct, nature and different from the indirect definition by taking the decomposable closure appearing in some references.Then we mainly explore the basic properties of the integral of fuzzy stochastic processes with respect to finite variation processes. We shall prove that for each α∈[0,1], the a-level set of the integral is closed, convex and the integral is a fuzzy stochastic process taking value in F(Rd). In addition, we shall show that the integrals of fuzzy process is L1-integrably bounded, at the same time, the integral is continuous in time t with respect to d∞ metric. Finally, we explore the representation theorem of integrals and two basic inequalities under the metric d∞.In the last, as an upcoming work, we will study the corresponding fuzzy stochastic differential equation. Under some suitable conditions, we may study the existence and uniqueness of the strong solution to the fuzzy stochastic differential equation. Besides, we also can discuss the continuity of strong solution under the d∞ metric.