Variance Gamma (non-local) equations

TL;DR

This paper introduces new non-local equations for the Variance Gamma process involving generalized Weyl derivatives, explores their connection to special functions, and studies convergence of compound Poisson processes to the Variance Gamma process.

Contribution

It presents novel non-local equations for the Variance Gamma process beyond the traditional time-changed Brownian motion framework.

Findings

Derived new non-local equations involving generalized Weyl derivatives.

Connected Variance Gamma equations to special functions.

Demonstrated convergence of compound Poisson processes to the Variance Gamma process.

Abstract

We provide some equations for the Variance Gamma process due to the fact that we do not consider only the definition as a time-changed Brownian motion. This brings us to a new non-local equation, even true in the drifted case, involving generalized Weyl derivatives. Then we focus on the connection to special functions and we study a space equation for our process. At the end, we conclude by observing the convergence in distribution of a compound Poisson process to the Variance Gamma process.