Creeping of L\'evy processes through curves

TL;DR

This paper investigates the probability that Lévy processes creep through curves, providing explicit formulas for bivariate subordinators and applications to real Lévy processes, including conditioned processes and Ornstein-Uhlenbeck models.

Contribution

It offers new explicit expressions for creeping probabilities of Lévy processes through curves, extending previous results to more general functions and process conditions.

Findings

Derived formulas for creeping probabilities of bivariate subordinators.

Applied results to real Lévy processes reaching their past supremum.

Provided examples including stable Ornstein-Uhlenbeck processes.

Abstract

A L\'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t,f(t)):t\ge0\}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L\'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L\'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.