Functional approximation of the marked Hawkes risk process

TL;DR

This paper introduces a discrete-time approximation for the continuous-time marked Hawkes risk process, proving convergence and providing bounds on the approximation error, which enhances modeling in fields like neuroscience, social networks, and insurance.

Contribution

It presents a novel strong discrete-time approximation of the Hawkes risk process with proven trajectorial convergence and explicit error bounds, extending existing theoretical results.

Findings

Proposed a discrete-time approximation of the Hawkes risk process.

Proved trajectorial convergence in Sobolev and Skorokhod spaces.

Derived explicit bounds on convergence speed depending on discretization and kernel regularity.

Abstract

The marked Hawkes risk process is a compound point process for which the occurrence and amplitude of past events impact the future. Thanks to its autoregressive properties, it found applications in various fields such as neuosciences, social networks and insurance.Since data in real life is acquired over a discrete time grid, we propose a strong discrete-time approximation of the continuous-time Hawkes risk process obtained be embedding from the same Poisson measure. We then prove trajectorial convergence results both in some fractional Sobolev spaces and in the Skorokhod space, hence extending the theorems proven in the literature. We also provide upper bounds on the convergence speed with explicit dependence on the size of the discretisation step, the time horizon and the regularity of the kernel.