×

zbMATH — the first resource for mathematics

Recurrent extensions of self-similar Markov processes and Cramér’s condition. (English) Zbl 1077.60055
Let \(\xi\) be a real-valued Lévy process that satisfies Cramér’s condition, and \(X\) a self-similar Markov process associated with \(\xi\) via Lamperti’s transformation. The author proves the existence of a unique excursion measure n, compatible with the semigroup \(X\) and such that \({\mathbf n}(X_{0+}>0)=0\). Some descriptions of the measure n are given.
Related paper: J. Lamperti [Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 205–225 (1972; Zbl 0274.60052)].

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G18 Self-similar stochastic processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press. · Zbl 0861.60003
[2] Bertoin, J. and Caballero, M.-E. (2002) Entrance from 0+ for increasing semi-stable Markov processes. Bernoulli, 8, 195-205. · Zbl 1002.60032
[3] Bertoin, J. and Doney, R.A. (1994) Cramérś estimate for Lévy processes. Statist. Probab. Lett., 21, 363-365. · Zbl 0809.60085
[4] Bertoin, J. and Yor, M. (2002a) The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal., 17(4), 389-400. · Zbl 1004.60046
[5] Bertoin, J. and Yor, M. (2002b) On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse, VI. Sér., Math., 11, 33-45. · Zbl 1031.60038
[6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989) Regular Variation. Cambridge: Cambridge University Press. · Zbl 0667.26003
[7] Blumenthal, R.M. (1983) On construction of Markov processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 63(4), 433-444. · Zbl 0494.60071
[8] Blumenthal, R.M. (1992) Excursions of Markov Processes. Boston: Birkhäuser. · Zbl 0983.60504
[9] Caballero, M. and Chaumont, L. (2004) Weak convergence of positive self similar Markov processes and overshoots of Lévy processes. Technical report, available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-899.pdf URL: · Zbl 1098.60038
[10] Carmona, P., Petit, F. and Yor, M. (1994) Sur les fonctionnelles exponentielles de certains processus de Lévy. Stochastics Stochastics Rep., 47(1-2), 71-101. · Zbl 0830.60072
[11] Carmona, P., Petit, F. and Yor, M. (1997) On the distribution and asymptotic results for exponential functionals of Lévy processes. In M. Yor (ed.), Exponential Functionals and Principal Values Related to Brownian Motion, Bibl. Rev. Mat. Iberoamericana, pp. 73-130. Madrid: Revista Matemática Iberoamericana. · Zbl 0905.60056
[12] Chaumont, L. (1997) Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math., 121(5), 377-403. · Zbl 0882.60074
[13] Dellacherie, C., Maisonneuve, B. and Meyer, P.A. (1992) Probabilités et Potentiel: Processus de
[14] Markov ( fin). Compléments du Calcul Stochastique, Vol. V. Paris: Hermann.
[15] Dufresne, D. (1990) The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J., 39-79. · Zbl 0743.62101
[16] Embrechts, P. and Maejima, M. (2002) Self-similar Processes. Princeton, NJ: Princeton University Press. · Zbl 1008.60003
[17] Fitzsimmons, P., Pitman, J. and Yor, M. (1993) Markovian bridges: construction, Palm interpretation, and splicing. In E. Çinlar, K.L. Chung and M.J. Sharpe (eds) Seminar on Stochastic Processes, 1992, Progr. Probab. 33, pp. 101-134. Boston: Birkhäuser. · Zbl 0844.60054
[18] Getoor, R.K. (1979) Excursions of a Markov process. Ann. Probab., 7(2), 244-266. · Zbl 0399.60069
[19] Getoor, R.K. and Sharpe, M.J. (1973) Last exit times and additive functionals. Ann. Probab., 1, 550- 569. · Zbl 0324.60062
[20] Getoor, R.K. and Sharpe, M.J. (1981) Two results on dual excursions, In E. Çinlar, K.L. Chung and R.K. Getoor (eds), Seminar on Stochastic Processes, 1981, Progr. Probab. Statist. 1, pp. 31-52. Boston: Birkhäuser. · Zbl 0531.60069
[21] Getoor, R.K. and Sharpe, M.J. (1982) Excursions of dual processes. Adv. in Math., 45(3), 259-309. · Zbl 0497.60067
[22] Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1(1), 126-166. · Zbl 0724.60076
[23] Imhof, J.-P. (1984) Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab., 21(3), 500-510. JSTOR: · Zbl 0547.60081
[24] Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math., 131, 207-248. · Zbl 0291.60029
[25] Lamperti, J. (1972) Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie Verw. Geb., 22, 205-225. · Zbl 0274.60052
[26] McKean, Jr., H.P. (1963) Excursions of a non-singular diffusion. Z. Wahrscheinlichkeitstheorie Verw. Geb., 1, 230-239. · Zbl 0117.35903
[27] Mejane, O. (2002) Cramerś estimate for the exponential functional of a Lévy process. Laboratoire de Statistiques et Probabilités, Université Paul Sabatier. Available at http://front.math.ucdavis.edu/math.PR/0211409 URL:
[28] Meyer, P.A. (1971) Processus de Poisson ponctuels, dáprès K. Ito. In Séminaire de Probabilités V, Lecture Notes in Math. 191, pp. 177-190. Berlin: Springer-Verlag.
[29] Mitro, J.B. (1984) Exit systems for dual Markov processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 66(2), 259-267. · Zbl 0525.60075
[30] Nagasawa, M. (1964) Time reversion of Markov processes. Nagoya Math. J., 24, 177-204. · Zbl 0133.10702
[31] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd edition. Berlin: Springer-Verlag. · Zbl 0917.60006
[32] Rivero, V.M. (2003) A law of iterated logarithm for increasing self-similar Markov processes. Stochastics Stochastics Rep., 75(6), 443-472. · Zbl 1053.60027
[33] Rogers, L.C.G. (1983) Itô excursion theory via resolvents. Z. Wahrscheinlichkeitstheorie Verw. Geb., 63(2), 237-255. · Zbl 0528.60073
[34] Salisbury, T.S. (1986a) Construction of right processes from excursions. Probab. Theory Related Fields, 73(3), 351-367. · Zbl 0587.60068
[35] Salisbury, T.S. (1986b) On the Itô excursion process. Probab. Theory Related Fields, 73(3), 319-350. · Zbl 0587.60067
[36] Sato, K.-I. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press. · Zbl 0973.60001
[37] Sharpe, M. (1988) General Theory of Markov Processes. Boston: Academic Press. · Zbl 0649.60079
[38] Vuolle-Apiala, J. (1994) Itô excursion theory for self-similar Markov processes. Ann. Probab., 22, 546-565. · Zbl 0810.60067
[39] Williams, D. (1974) Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3), 28, 738-768. · Zbl 0326.60093
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.