Summary
We study the behaviour of a Lévy process with no positive jumps near its increase times. Specifically, we construct a local time on the set of increase times. Then, we describe the path decomposition at an increase time chosen at random according to the local time, and we evaluate the rate of escape before and after this instant.
References
Bertoin, J.: Increase of a Lévy process with no positive jumps. Stochastics37, 247–251 (1991)
Bertoin, J.: Sur la décomposition de la trajectoire d'un processus de Lévy spectralement positif en son infimum. Ann. Inst. H. Poincaré27-4, 537–547 (1991)
Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab.7, 705–766 (1975)
Chung, K.L., Walsh, J.B.: To reverse a Markov process. Acta Math.123, 225–251 (1969)
Doney, R.A.: Hitting probabilities for spectrally positive Lévy processes. J. Lond. Math. Soc.44, 566–576 (1991)
Dvoretzky, A., Erdös, P., Kakutani, S.: On nonincrease everywhere, of the Brownian motion process. In: Proc. 4th. Berkeley Symp. Math. Stat. and Probab. II, pp. 103–116. Berkeley, CA: University of California Press 1961
Fitzsimmons, P.J., Fristedt, B., Maisonneuve, B.: Intersections and limits of regenerative sets. Z. Wahrscheinlichkeitstheor Verw. Geb.70, 157–173 (1985)
Fristedt, B.: Sample function behaviour of increasing processes with stationary independent increments. Pac. J. Math.21, 21–33 (1967)
Fristedt, B., Pruitt, W.: Lower functions for increasing random walks and subordinators. Z. Wahrscheinlichkeitstheor. Verw. Geb.18, 167–182 (1971)
Greenwood, P., Pitman, J.: Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab.12, 893–902 (1980)
Millar, P.W.: Exit properties of stochastic processes with stationary independent increments. Trans. Am. Math. Soc.178, 459–479 (1973)
Millar, P.W.: Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc.226, 365–391 (1977)
Monrad, D., Silverstein, M.L.: Stable processes: Sample function growth at a local minimum. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 177–210 (1979)
Takàcs, L.: Combinatorial methods in the theory of stochastic processes. New-York: Wiley 1967
Taylor, S.J., Wendel, J.G.: The exact Hausdorff measure of the zero-set of a stable process. Z. Wahrscheinlichkeitstheor. Verw. Geb.6, 170–180 (1966)
Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions. Proc. Lond. Math. Soc.28, 738–768 (1974)
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Bertoin, J. Lévy processes with no positive jumps at an increase time. Probab. Th. Rel. Fields 96, 123–135 (1993). https://doi.org/10.1007/BF01195886
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DOI: https://doi.org/10.1007/BF01195886
Mathematics Subject Classification (1980)
- 60 J 30
- 60 G 17