Subexponential potential asymptotics with applications

TL;DR

This paper studies the asymptotic behavior of potential functions for a multivariate process with small jumps and heavy-tailed large jumps, relevant for insurance risk models, using renewal equations.

Contribution

It introduces a novel analysis of potential asymptotics for processes with subexponential jumps, extending understanding of heavy-tailed risk models.

Findings

Asymptotic behavior characterized by subexponential tail properties

Renewal equation approach for potential functions

Application to insurance claim modeling

Abstract

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$, $X_0=x$, killed at some terminal time $T$, where $Y_t$ is a Markov process having only jumps of the length smaller than $\delta$, and $Z_t$ is a compound Poisson process with jumps of the length bigger than $\delta$ for some fixed $\delta>0$. Under the assumptions that the summands in $Z_t$ are sub-exponential, we investigate the asymptotic behaviour of the potential function $u(x)= E^x \int_0^\infty \ell(X_s^\sharp)ds$. The case of heavy-tailed entries in $Z_t$ corresponds to the case of "big claims" in insurance models and is of practical interest. The main approach is based on fact that $u(x)$ satisfies a certain renewal equation.