Lévy processes conditioned on having a large height process

Richard, Mathieu

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

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.

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.

[1] R. Abraham and J.-F. Delmas. Feller property and infinitesimal generator of the exploration process. J. Theoret. Probab. 20 (2007) 355-370. | MR

[2] D. Aldous. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (1991) 228-266. | MR

[3] K. B. Athreya and P. E. Ney. Branching Processes. Grundlehren der mathematischen Wissenschaften 196. Springer, New York, 1972. | MR

[4] J. Azéma and M. Yor. Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) 90-115. Lecture Notes in Math. 721. Springer, Berlin, 1979. | Numdam | MR

[5] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996. | MR

[6] L. Chaumont. Sur certains processus de Lévy conditionnés à rester positifs. Stochastics Stochastics Rep. 47 (1994) 1-20. | MR

[7] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR

[8] L. Chaumont and R. A. Doney. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005) 948-961 (electronic). | MR

[9] T. Duquesne. Continuum random trees and branching processes with immigration. Stochastic Process. Appl. 119 (2009) 99-129. | MR

[10] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. | Numdam | MR

[11] J. Geiger. Size-biased and conditioned random splitting trees. Stochastic Process. Appl. 65 (1996) 187-207. | MR

[12] J. Geiger. Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab. 36 (1999) 301-309. | MR

[13] J. Geiger and G. Kersting. Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 111-126. IMA Vol. Math. Appl. 84. Springer, New York, 1997. | MR

[14] K. Hirano. Lévy processes with negative drift conditioned to stay positive. Tokyo J. Math. 24 (2001) 291-308. | MR

[15] D. P. Kennedy. Some martingales related to cumulative sum tests and single-server queues. Stochastic Processes Appl. 4 (1976) 261-269. | MR

[16] A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 (2002) 42-70. | MR

[17] A. Lambert. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007) 420-446. | MR

[18] A. Lambert. Population dynamics and random genealogies. Stoch. Models 24 (2008) 45-163. | MR

[19] A. Lambert. The contour of splitting trees is a Lévy process. Ann. Probab. 38 (2010) 348-395. | MR

[20] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR

[21] Z.-H. Li. Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 (2000) 68-84. | MR

[22] V. Limic. A LIFO queue in heavy traffic. Ann. Appl. Probab. 11 (2001) 301-331. | MR

[23] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of $LlogL$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125-1138. | MR

[24] P. Millar. Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc. 226 (1977) 365-391. | MR

[25] L. Nguyen-Ngoc. Limiting laws and penalization of certain Lévy processes by a function of their maximum. Teor. Veroyatn. Primen. 55 (2010) 530-547. | MR

[26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der mathematischen Wissenschaften 293. Springer, Berlin, 1999. | MR

[27] P. Robert. Stochastic Networks and Queues, french edition. Applications of Mathematics (New York) 52. Stochastic Modelling and Applied Probability. Springer, Berlin, 2003. | MR

[28] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795-798. | MR