Exponential martingales and changes of measure for counting processes

TL;DR

This paper establishes practical criteria for the Doléans-Dade exponential of stochastic integrals with respect to counting process martingales to be true martingales, enabling the construction of complex counting processes like Hawkes processes.

Contribution

It provides new, weak, and verifiable conditions specifically tailored for counting processes to ensure the exponential martingale property.

Findings

Criteria applicable to nonexplosive Hawkes processes

Conditions for counting processes with stochastic intensities

Examples demonstrating the criteria's usefulness

Abstract

We give sufficient criteria for the Dol\'eans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes as well as counting processes with stochastic intensities depending on diffusion processes.