Levy processes: long time behavior and convolution-type form of the Ito representation of the infinitesimal generator

TL;DR

This paper reformulates the Levy-Ito representation of the infinitesimal generator in a convolution form, constructs a quasi-potential operator, and derives a new formula to analyze the long-term behavior of the process within a domain.

Contribution

It introduces a convolution-type form of the Levy-Ito representation and constructs a quasi-potential operator to study long-term probabilities of Levy processes.

Findings

Derived a convolution form of the Levy-Ito generator

Constructed a quasi-potential operator for Levy processes

Obtained a new formula for the ruin probability p(t,Δ)

Abstract

In the present paper we show that the Levy-Ito representation of the infinitesimal generator $L$ for Levy processes $X_t$ can be written in a convolution-type form. Using the obtained convolution form we have constructed the quasi-potential operator $B$. We denote by $p(t,\Delta)$ the probability that a sample of the process $X_t$ remains inside the domain $\Delta$ for $0{\leq}\tau{\leq}t$ (ruin problem). With the help of the operator $B$ we find a new formula for $p(t,\Delta)$. This formula allows us to obtain long time behavior of $p(t,\Delta)$.