Self-exciting jump processes and their asymptotic behaviour

Abstract

The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite variation stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that for each fixed point in time, t≥0, a scaling limit of the intensity process converges in distribution, and the limit equals the strong solution of the square-root diffusion process (Cox–Ingersoll–Ross process) at t. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case.