Stochastic calculus for convoluted L\'{e}vy processes

TL;DR

This paper introduces an elementary stochastic calculus framework for convoluted Lévy processes, including fractional Lévy processes, using a Skorokhod integral and an Itô formula that accounts for memory and jumps.

Contribution

It develops a new stochastic calculus for convoluted Lévy processes without relying on Malliavin calculus, providing an elementary approach and an Itô formula.

Findings

Derived an Itô formula separating memory and jump contributions.

Established an elementary Skorokhod integral for convoluted Lévy processes.

Included fractional Lévy processes within the calculus framework.

Abstract

We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps.