Convolution type form of the Ito representation of the infinitesimal generator for Levy processes

TL;DR

This paper demonstrates that the Ito representation of the infinitesimal generator for Levy processes can be expressed as a convolution, enabling new analysis of Levy process properties through integral equations.

Contribution

It introduces a convolution form of the Ito representation for Levy processes' generators, providing a novel approach to studying their properties.

Findings

Convolution form of the Ito generator for Levy processes established.

Analysis of Levy process properties via integral equations developed.

Enhanced understanding of Levy processes through convolution representation.

Abstract

In the present paper we show that the Ito representation of the infinitesimal generator $L$ for Levy processes can be written in a convolution type form. Using the obtained convolution form and the theory of integral equations with difference kernels we study the properties of Levy processes.