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Completely asymmetric stable processes conditioned to avoid an interval

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  11 December 2019

Pierre Lenthe  and
Philip Weissmann
Pierre Lenthe*
Affiliation:
University of Mannheim
Philip Weissmann*
Affiliation:
University of Mannheim
*
*Postal address: Institute of Mathematics, University of Mannheim, 68161 Mannheim,Germany.
*Postal address: Institute of Mathematics, University of Mannheim, 68161 Mannheim,Germany.
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Abstract

In a recent article, Döring et al. (2018) conditioned a stable process with two-sided jumps to avoid an interval. As usual, the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found by Döring et al. does not exist or is not invariant, but nonetheless we will characterise the conditioned process as a Markov process.

Keywords

Markov processLévy processstable processDoob h-transform

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60J45: Probabilistic potential theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1187 - 1197
Copyright
© Applied Probability Trust 2019 

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References

Bertoin, J. (1996). Lévy Processes, Vol. 121 of Cambridge Tracts in Mathematics. Cambridge University Press.Google Scholar
Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Prob. 10, 948–961.CrossRefGoogle Scholar
Chung, K. and Walsh, J. (2005). Markov Processes, Brownian Motion, and Time Symmetry. Grundlehren der mathematischen Wissenschaften. Springer, New York.CrossRefGoogle Scholar
Döring, L., Kyprianou, A. E. and Weissmann, P. (2018). Stable processes conditioned to avoid an interval. Preprint, arXiv:1802.07223.Google Scholar
Döring, L., Watson, A. R. and Weissmann, P. (2019). Lévy processes with finite variance conditioned to avoid an interval. Electron. J. Prob. 24, 1–32.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. d’Analyse Mathématique 20, 371–395.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2004). Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. Springer, Berlin.Google Scholar
Rogozin, B. A. (1973). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Prob. Appl. 17, 332–338.CrossRefGoogle Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Vol. 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google Scholar