Markov approximation for controlled Hawkes Jump-Diffusions with general kernels

TL;DR

This paper introduces a Markov approximation method for controlled Hawkes jump-diffusions with general kernels, enabling their approximation by Markov jump-diffusions and facilitating applications in stochastic control.

Contribution

It proposes a novel approximation technique for Hawkes jump-diffusions with general kernels using exponential functions, extending their tractability in stochastic control.

Findings

Hawkes kernels can be approximated by linear combinations of exponentials.

Hawkes jump-diffusions can be effectively approximated by Markov jump-diffusions.

Application demonstrated in stochastic control problems.

Abstract

We present a Markov approximation for jump-diffusions whose jump part consists in a Hawkes process with intensity driven by a general (possibly non-monotone) kernel. Under minimal integrability conditions, the kernel can be approximated by a linear combination of exponential functions. This implies that Hawkes jump-diffusions can be approximated with Markov jump-diffusions. We illustrate the usefulness of this approximation by applying it to a class of stochastic control problems.