Self-exciting jump processes and their asymptotic behaviour

TL;DR

This paper investigates the properties and asymptotic behaviour of self-exciting jump processes, deriving formulas for moments, expectations, variances, and their scaling limits, including connections to the Cox-Ingersoll-Ross process.

Contribution

It introduces recursive formulas for moments of self-exciting processes and establishes their asymptotic behaviour, linking the intensity process to the CIR model.

Findings

Self-exciting processes can exhibit finite or infinite activity.

Explicit formulas for expectation and variance in linear intensity cases.

Scaling limits of the intensity process relate to the CIR process.

Abstract

The purpose of this paper is to investigate properties of self-exciting jump processes. We derive the Laplace transform of SDE driven self-exciting processes with independent, identically distributed jump sizes. By using this Laplace transform, we find a recursive formula for the moments of the self-exciting process. The formula for the moments allow us to derive expressions for the expectation and variance of the self-exciting process. We show that self-exciting processes can exhibit both finite and infinite activity behaviour. Furthermore, we show that the scaling limit of the intensity process equals the strong solution of the square-root diffusion process(Cox-Ingersoll-Ross process) in distribution. 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.