2011/06/11 by Mathieu Richard, Richard, Mathieu
Mathematics · #60G07 #60G44 #60G51 #60G57 (Secondary) #60J80 (Primary) 60J85 #60K25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60G07 #msc:60G44 #msc:60G51 #msc:60G57 #msc:60J80 #msc:60J85 #msc:60K25
paper · pdf · doi:10.48550/arxiv.1106.2245
34 pages, 2 figures
openalex publication_date 2011/06/11 · arxiv created 2012/03/20 · arxiv updated 2012/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present work, we consider spectrally positive Lévy processes (Xt,t≥0) not drifting to +∞ 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 \pfl of this conditioned process is defined as a Doob h-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is (∫\rt(dz)eαz+It)\2t≤ T0 for some α and where (It,t≥0) is the past infimum process of X, where (\rt,t≥0) is the so-called exploration process defined in Duquesne, 2002, and where T0 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→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.