×

Increase of Lévy processes. (English) Zbl 0862.60027

Summary: A rather complicated condition is shown to be necessary and sufficient for a Lévy process to have points of increase. A much simpler condition is then shown to be sufficient in the general case, and necessary under certain regularity conditions. The approach used here also gives a unified proof of results for certain special classes of Lévy processes, which have previously been obtained by Bertoin.

MSC:

60G17 Sample path properties
60J99 Markov processes
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adelman, O. (1985). Brownian motion never increases: a new proof to a result of Dvoretzky, Erdös and Kakutani. Israel J. Math. 50 189-192. · Zbl 0573.60030
[2] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York. · Zbl 0679.60013
[3] Bertoin, J. (1991). Increase of a Lévy process with no positive jumps. Stochastics 37 247-251. · Zbl 0739.60065
[4] Bertoin, J. (1993). Lévy processes with no positive jumps at an increase time. Probab. Theory Related Fields 96 123-135. · Zbl 0794.60075
[5] Bertoin, J. (1994). Increase of stable processs. J. Theoret. Probab. 7 551-563. · Zbl 0809.60050
[6] Bertoin, J. (1994). Lévy processes that can creep downwards never increase. Ann. Inst. H. Poincaré Probab. Statist. 31 379-391. · Zbl 0816.60073
[7] Bertoin, J. (1996). An Introduction to Lévy Processes. Cambridge Univ. Press.
[8] Bertoin, J. and Doney, R. A. (1994). Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 363-365. · Zbl 0809.60085
[9] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705-766. JSTOR: · Zbl 0322.60068
[10] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001
[11] Burdzy, K. (1990). On nonincrease of Brownian motion. Ann. Probab. 18 978-980. · Zbl 0719.60086
[12] Dvoretzky, A., Erd ös, P. and Kakutani, S. (1961). Nonincrease every where of the Brownian motion process. Fourth Berkeley Sy mp. Math. Statist. Probab. 2 103-116. Univ. California Press, Berkeley. · Zbl 0111.15002
[13] Fristedt, B. E. (1973). Sample functions of stochastic processes with stationary independent increments. Adv. Probab. 3 241-396. · Zbl 0309.60047
[14] Greenwood, P. E. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. in Appl. Probab. 12 893-902. · Zbl 0443.60037
[15] Horowitz, J. (1972). Semilinear Markov process, subordinators and renewal theory. Z. Wahrsch. Verw. Gebiete 24 167-193. · Zbl 0251.60052
[16] Kesten, H. (1969). Hitting probabilities of single points for processes with stationary independent increments. Mem. Amer. Math. Soc. 93 1-129. · Zbl 0201.19002
[17] Knight, F. B. (1981). The Essentials of Brownian Motion and Diffusion. Math. Survey s 18. Amer. Math. Soc., Providence, RI. · Zbl 0458.60002
[18] Millar, P. W. (1973). Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc. 178 459-479. · Zbl 0268.60065
[19] Rogers, L. C. J. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 20 21-34.
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.