Lévy processes conditioned on having a large height process

Richard, Mathieu

Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 982-1013.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ 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 Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $x>0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int {\tilde{\rho }}_t\left(\mathrm{d}z\right){\mathrm{e}}^{\alpha z}+I_t){\mathbf{1}}_{\{t\le T_0\}}$ for some $\alpha$ and where $(I_t,t\ge 0)$ is the past infimum process of $X$, where $({\tilde{\rho }}_t,t\ge 0)$ is the so-called exploration process defined in [10] and where $T_0$ is the hitting time of 0 for $X$. Under ${\mathbb{P}}_x^{☆}$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of ${\mathbb{P}}_x^{☆}$ as $x\to 0$. 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 splitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

Dans ce travail, on considère des processus de Lévy $(X_t,t\ge 0)$ ne dérivant pas vers $+\infty$ et on s’intéresse à leur conditionnement à atteindre des hauteurs arbitrairement grandes (au sens du processus des hauteurs associé à $X$) avant de toucher $0$. On obtient ainsi une nouvelle manière de conditionner des processus de Lévy à rester positifs. La loi (honnête) ${\mathbb{P}}_x^{☆}$ de ce processus conditionné (partant de $x>0$) est définie selon une $h$-transformée de Doob à l’aide d’une martingale. En ce qui concerne les processus de Lévy ayant des trajectoires à variation infinie, cette martingale est $(\int {\tilde{\rho }}_t\left(\mathrm{d}z\right){\mathrm{e}}^{\alpha z}+I_t){\mathbf{1}}_{\{t\le T_0\}}$ pour un certain $\alpha$ et où $(I_t,t\ge 0)$ est le processus infimum de $X$, où $({\tilde{\rho }}_t,t\ge 0)$ est le processus d'exploration défini dans [10] et où $T_0$ est le temps d’atteinte de 0 par $X$. Sous ${\mathbb{P}}_x^{☆}$, on obtient également une décomposition de la trajectoire de $X$ en son minimum; ce qui permet de prouver la convergence de ${\mathbb{P}}_x^{☆}$ quand $x\to 0$. Lorsque le processus $X$ est un processus de Poisson composé compensé, la martingale est définie à partir des sauts du processus infimum futur de $X$. Les preuves sont plus simples dans ce cas puisque on peut voir $X$ comme le processus de contour d’un arbre de ramification (sous)critique. Dans ce cas, on énonce aussi une caractérisation alternative du processus conditionné dans l'esprit des décompositions spinales.

Classification: 60G51, 60J80, 60J85, 60G44, 60K25, 60G07, 60G57
Keywords: Lévy process, height process, Doob harmonic transform, splitting tree, spine decomposition, Size-biased distribution, queueing theory

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