Study of Stochastic Differential Equation Driven by Time-Changed Levy Noise

Abstract

This dissertation is composed of two parts. The first part studies stabilities of the solution of stochastic differential equation (SDE) driven by time-changed L´evy noise in probability, moment, and path sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for solution of time-changed SDE to be stable in different senses will be established. Connection between stability of solution to time-changed SDE and that to corresponding original SDE will be disclosed. The second part studies a time-changed stochastic control problem, where the underlying stochastic process is a L´evy noise time-changed by an inverse subordinator. We establish a maximum principle theory for the time-changed stochastic control problem. We also prove the existence and uniqueness of the corresponding time-changed backward stochastic differential equation involved in the stochastic control problem. Some examples are provided for illustration.