L\'evy processes conditioned on having a large height process

TL;DR

This paper introduces a new way to condition spectrally positive Lévy processes to reach large heights before hitting zero, using a Doob h-transform, and explores their path decompositions and convergence properties.

Contribution

It develops a novel conditioning method for Lévy processes based on height processes, including explicit martingales and path decompositions, extending previous work on splitting trees and spine decompositions.

Findings

Defined a new conditioned law via Doob h-transform.

Established path decomposition at the process minimum.

Proved convergence of the conditioned process as initial height tends to zero.

Abstract

In the present work, we consider spectrally positive L\'evy processes $(X_t,t\geq0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting 0. This way we obtain a new conditioning of L\'evy processes to stay positive. The (honest) law $\pfl$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For L\'evy processes with infinite variation paths, this martingale is $(\int\tilde\rt(\mathrm{d}z)e^{\alpha z}+I_t)\2{t\leq T_0}$ for some $\alpha$ and where $(I_t,t\geq0)$ is the past infimum process of $X$, where $(\tilde\rt,t\geq0)$ is the so-called \emph{exploration process} defined in Duquesne, 2002, and where $T_0$ is the hitting time of 0 for $X$. Under $\pfl$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\pfl$ as $x\to0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)critical \emph{splitting tree}. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.