Simulation of stochastic Volterra equations driven by space--time L\'evy noise

TL;DR

This paper develops and analyzes two numerical schemes for simulating stochastic Volterra equations driven by space--time Lévy noise, providing convergence proofs, explicit rates, and simulation visualizations.

Contribution

It introduces and compares two novel numerical methods for stochastic Volterra equations with Lévy noise, including convergence analysis and explicit error rates.

Findings

Both methods converge in $L^p$ and almost surely.

Explicit convergence rates depend on kernel and noise characteristics.

Simulation results illustrate key path properties.

Abstract

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space--time L\'evy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods $L^p$- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.