On subexponential tails for the maxima of negatively driven compound renewal and L\'evy processes

TL;DR

This paper investigates the tail behavior of the maximum of certain stochastic processes with negative drift, including compound renewal and Lévy processes, establishing a principle of a single big jump for their maxima across all time horizons.

Contribution

It provides new subexponential tail asymptotics for the maxima of these processes and proves the principle of a single big jump applicable to both classes.

Findings

Subexponential tail asymptotics for process maxima.

Principle of a single big jump established.

Applicable to compound renewal and Lévy processes.

Abstract

We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and L\'evy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cram\'er-Lundberg risk process.