Malliavin calculcus for a Hawkes process

TL;DR

This paper develops a Malliavin calculus framework for nonlinear Hawkes processes, enabling analysis of their properties and applications in financial derivative valuation.

Contribution

It introduces a novel Malliavin calculus approach for nonlinear Hawkes processes based on jump time perturbations, leading to new analytical tools.

Findings

Established criteria for absolute continuity of solutions to SDEs driven by Hawkes processes.

Derived sensitivity formulas for financial derivatives with respect to model parameters.

Constructed a local Dirichlet form for nonlinear Hawkes processes.

Abstract

We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.