Set-valued Stochastic Integral With Respect To Bounded Variation Rrocesses

This paper studies the Lebesgue-Stieltjes integrals of set-valued stochastic processes with respect to a real valued bounded variation process and the corresponding set-valued stochastic differential equations.This thesis consists of three main parts.The first part is mainly on the Lebesgue-Stieltjes integrals of set-valued stochastic processes with respect to monotone increasing function, which are defined directly by employing all integrably selections instead of taking the decomposable closure appearing in some existed references. For every time t, we shall show that this kind of integral is a set-valued stochastic variable taking values in the power set K(Rd). Furthermore, the process of integrals is jointly measurable in the product space and L1-integrably bounded. We also prove its representation theorem. It is also continuous in time t with respect to the Hausdorff metric.The second part studies the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to monotone non-increasing process and studies some properties of it. We show the integral is measurable. Furthermore, it is measurable with respect to the joint a algebra and L1-integrably bounded. It is also continuous in time t with respect to the Hausdorff metric.The third part introduces the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to real valued bounded variation process. We still employ all integrably bounded selections instead of taking the decomposable closure to define the Lebesgue-Stieltjes integral. We also show that this kind of integral is measurable, continuous in t under the Hausdorff metric and L1-bounded. Then we discuss the existence and uniqueness of strong solution to the set-valued stochastic differential equation with the Lebesgue-Stieltjes set-valued integrals. Also we discuss some properties of solutions such as the continuity of t under the Hausdorff metric.