×

zbMATH — the first resource for mathematics

Recurrent extensions of self-similar Markov processes and Cramér’s condition. II. (English) Zbl 1132.60056
Summary: We prove that a positive self-similar Markov process \((X, \mathbb P)\) that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition.
For part I, see ibid. 11, No. 3, 471–509 (2005; Zbl 1077.60055).

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G18 Self-similar stochastic processes
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Bertoin, J. and Caballero, M.-E. (2002). Entrance from \(0+\) for increasing semi-stable Markov processes., Bernoulli 8 195–205. · Zbl 1002.60032
[2] Bertoin, J. and Doney, R.A. (1994). Cramér’s estimate for Lévy processes., Statist. Probab. Lett. 21 363–365. · Zbl 0809.60085
[3] Bertoin, J. and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes., Potential Anal. 17 389–400. · Zbl 1004.60046
[4] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes., Probab. Surv. 2 191–212. · Zbl 1189.60096
[5] Blumenthal, R.M. (1983). On construction of Markov processes., Z. Wahrsch. Verw. Gebiete 63 433–444. · Zbl 0494.60071
[6] Blumenthal, R.M. (1992)., Excursions of Markov Processes . Probability and Its Applications. Boston: Birkhäuser. · Zbl 0983.60504
[7] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In, Exponential Functionals and Principal Values Related to Brownian Motion . Bibl. Rev. Mat. Iberoamericana 73–130. Madrid: Rev. Mat. Iberoamericana. · Zbl 0905.60056
[8] Dellacherie, C., Maisonneuve, B. and Meyer, P.-A. (1992)., Probabilités et potentiel: Processus de Markov (fin). Compléments du Calcul Stochastique . Paris: Hermann.
[9] Erickson, K.B. (1970). Strong renewal theorems with infinite mean., Trans. Amer. Math. Soc. 151 263–291. JSTOR: · Zbl 0212.51601
[10] Fitzsimmons, P. (2006). On the existence of recurrent extensions of self-similar Markov processes., Electron. Comm. Probab. 11 230–241 (electronic). · Zbl 1110.60036
[11] Getoor, R.K. (1990)., Excessive Measures . Probability and Its Applications. Boston: Birkhäuser. · Zbl 0982.31500
[12] Getoor, R.K. and Sharpe, M.J. (1973). Last exit times and additive functionals., Ann. Probab. 1 550–569. · Zbl 0324.60062
[13] Goldie, C.M. (1991). Implicit renewal theory and tails of solutions of random equations., Ann. Appl. Probab. 1 126–166. · Zbl 0724.60076
[14] Lamperti, J. (1972). Semi-stable Markov processes. I., Z. Wahrsch. Verw. Gebiete 22 205–225. · Zbl 0274.60052
[15] Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes., Stochastic Process. Appl. 116 156–177. · Zbl 1090.60046
[16] Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér’s condition., Bernoulli 11 471–509. · Zbl 1077.60055
[17] Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes., Ann. Probab. 22 546–565. · Zbl 0810.60067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.