Compound Multivariate Hawkes Processes: Large Deviations and Rare Event Simulation

TL;DR

This paper develops a large deviations framework for multivariate compound processes driven by Hawkes processes with random marks, enabling analysis of rare events and efficient simulation methods.

Contribution

It establishes a large deviations principle for multivariate compound Hawkes processes and introduces an asymptotically efficient importance sampling method for rare event simulation.

Findings

Proved large deviations principle for the process.

Derived asymptotic ruin probabilities.

Demonstrated efficiency of the importance sampling method.

Abstract

In this paper, we establish a large deviations principle for a multivariate compound process induced by a multivariate Hawkes process with random marks. Our proof hinges on showing essential smoothness of the limiting cumulant of the multivariate compound process, resolving the inherent complication that this cumulant is implicitly characterized through a fixed-point representation. We employ the large deviations principle to derive logarithmic asymptotic results on the marginal ruin probabilities of the associated multivariate risk process. We also show how to conduct rare event simulation in this multivariate setting using importance sampling and prove the asymptotic efficiency of our importance sampling based estimator. The paper is concluded with a systematic assessment of the performance of our rare event simulation procedure.