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Conditionings and path decompositions for Lévy processes. (English) Zbl 0879.60072

Summary: We first give an interpretation for the conditioning to stay positive respectively, to die at 0 for a large class of Lévy processes starting at \(x >0\). Next, we specify the laws of the pre-minimum and post-minimum parts of a Lévy process conditioned to stay positive. We show that these parts are independent and have the same law as the process conditioned to die at 0 and the process conditioned to stay positive starting at 0, respectively. Finally, in some special cases, we prove the Skorohod convergence of this family of laws when \(x\) goes to 0.

MSC:

60J25 Continuous-time Markov processes on general state spaces
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[1] Bertoin, J., Sur la décomposition de la trajectoire d’un processus de Lévy spectralement positif en son infimum, Ann. Inst. H. Poincaré, 27, 537-547 (1991) · Zbl 0758.60073
[2] Bertoin, J., An extension of Pitman’s theorem for spectrally positive Lévy processes, Ann. Probab., 20, 1464-1483 (1992) · Zbl 0760.60068
[3] Bertoin, J., Splitting at the infimum and excursions in half-lines for random walks and Lévy processes, Stoch. Process. Appl., 47, 17-35 (1993) · Zbl 0786.60101
[4] Bertoin, J., Increase of stable processes, J. Theoret. Probab., 551-563 (1994) · Zbl 0809.60050
[5] Bertoin, J., Lévy Processes (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0861.60003
[6] Bertoin, J.; Doney, R. A., On conditioning a random walk to stay nonnegative, Ann. Probab., 22, 2152-2167 (1994) · Zbl 0834.60079
[7] Bingham, N., Maxima of sums of random variables and suprema of stable processes, Z. Wahrscheinlichkeitstheorie verw. Geb., 26, 273-296 (1973) · Zbl 0238.60036
[8] Chaumont, L., Sur certains processus de Lévy conditionnés à rester positifs, Stochastics, Stochastics Reports, 47, 1-20 (1994) · Zbl 0827.60064
[9] Chaumont, L., Processus de Lévy et conditionnement, Thèse de Doctorat de l’université (1994), Paris VI · Zbl 0827.60064
[10] Chaumont, L., Excursion normalisée, méandre et pont pour des processus stables, Bull. Sci. Math. (1996), to appear · Zbl 0882.60074
[11] Dellacherie, C.; Maisonneuve, B.; Meyer, P. A., (Probabilités et potentiel, Vol. 5 (1992), Hermann: Hermann Paris)
[12] Dellacherie, C.; Meyer, P. A., (Probabilités et potentiel, Vol. 4 (1987), Hermann: Hermann Paris)
[13] Greenwood, P.; Pitman, J. W., Fluctuation identities for Lévy processes and splitting at the Maximum, Adv. Appl. Probab., 12, 893-902 (1980) · Zbl 0443.60037
[14] Kesten, H., Hitting probability of single points for processes with stationary independent increments, Mem. Amer. Math. Soc., 93 (1969) · Zbl 0201.19002
[15] Maisonneuve, B., Processus de Markov: naissance, retournement, régénération, (Ecole d’été de Probabilités de Saint-Flour XXI. Ecole d’été de Probabilités de Saint-Flour XXI, Lecture Notes in Mathematics, Vol. 1541 (1974), Springer: Springer Berlin), 261-292 · Zbl 0844.60042
[16] marsalle, L., Hausdorff measures and capacities for increase times of stable processes (1995), Prépublication du Laboratoire de Probabilités de l’université Paris VI · Zbl 0918.60070
[17] Millar, P. W., Exit properties of stochastic processes with stationary independent increments, Trans. Amer. Math. Soc., 178, 459-479 (1973) · Zbl 0268.60065
[18] Millar, P. W., Zero-one laws and the minimum of a Markov process, Trans. Amer. Math. Soc., 226, 365-391 (1977) · Zbl 0381.60062
[19] Nagasawa, M., Time reversion of Markov processes, Nagoya Math. J., 24, 117-204 (1964) · Zbl 0133.10702
[20] Silverstein, M. L., Classification of coharmonic and coinvariant functions for a Lévy process, Ann. Probab., 8, 539-575 (1980) · Zbl 0459.60063
[21] Skohorod, A. V., Limit theorems for stochastic processes with independent increments, Theory. Probab. Appl., 2, 138-171 (1957)
[22] Williams, D., Path decomposition and continuity of local time for one dimentional diffusions, (Proc. London Math. Soc., 28 (1974)), 738-768 · Zbl 0326.60093
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