Weixuan Xia, USC

Title: Set-Valued Stochastic Integrals and Convoluted Lévy Processes

In this talk, I will discuss set-valued Volterra-type stochastic integrals driven by Lévy processes. I will explain the definition of set-valued convoluted stochastic integrals by taking the closed decomposable hull of integral functionals over time, thereby extending classical definitions to convoluted integrals with square-integrable kernels. Two key insights include: (1) Aside from established results for set-valued Itô integrals, while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are guaranteed to be integrally bounded for bounded integrands, this is not true when the random measure represents infinite variation; (2) It is a mutual effect of kernel singularity and jumps that the set-valued convoluted integrals are possibly explosive and can take extended vector values. These findings carry significant implications for the construction of set-valued fractional dynamical systems. Additionally, I will explore two classes of set-monotone processes for practical interests in economic and financial modeling.