On stochastic integration for volatility modulated L\'{e}vy-driven Volterra processes

TL;DR

This paper develops a stochastic integration framework for volatility modulated Lévy-driven Volterra processes, incorporating stochastic volatility and jumps, using Malliavin calculus to define an anticipative integral with key properties and applications.

Contribution

It introduces a new stochastic integral for VMLV processes that accounts for stochastic volatility and jumps, extending existing theories.

Findings

Defines an anticipative stochastic integral using Malliavin calculus.

Establishes fundamental properties of the new integral.

Provides applications demonstrating the theory's usefulness.

Abstract

This papers develops a stochastic integration theory with respect to volatility modulated L\'{e}vy-driven Volterra (VMLV) processes. It extends recent results in the literature to allow for stochastic volatility and pure jump processes in the integrator. The new integration operator is based on Malliavin calculus and describes an anticipative integral. Fundamental properties of the integral are derived and important applications are given.