On the distribution of a rotationally invariant α-stable process at the hitting time of a given hyperplane

Authors

  • M.M. Osypchuk Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk
  • M.I. Portenko Institute of Mathematics of the NAS of Ukraine, Kiev

DOI:

https://doi.org/10.15407/dopovidi2018.12.014

Keywords:

hitting time, local time, resolvent kernel, α-stable process

Abstract

We find out an explicit formula for the distribution of a rotationally invariant α-stable process at that moment of time, when it hits a given hyperplane for the first time. The case of 1<α⩽2 is considered.

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References

Dynkin, E. B. (1965). Markov Processes. (Vols. 1, 2). New York: Academic Press; Berlin: Springer. doi: https://doi.org/10.1007/978-3-662-00031-1

Blumenthal, R. M., Getoor, R. K. & Ray, D. B. (1961). On the distribution of first hits for the symmetric stable process. Trans. Am. Math. Soc., 99, No. 3, pp. 540-554.

Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Anal. Math., 20, pp. 371-395. doi: https://doi.org/10.1007/BF02786681

Rogozin, B. A. (1971). The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Probab. Appl., 16, pp. 575-595. doi: https://doi.org/10.1137/1116067

Caballero, M. E., Pardo, J. C. & Pérez, J. L. (2011). Explicit identities for Lévy process associated to symmetric stable processes. Bernoulli, 17, pp. 34-59. doi: https://doi.org/10.3150/10-BEJ275

Kyprianou, A. E., Pardo, J. C. & Watson, A. R. (2014). Hitting distributions of α-stable processes via path censoring and self-similarity. Ann. Probab., 42, No. 1, pp. 398-430. doi: https://doi.org/10.1214/12-AOP790

Bochner, S. (1959). Lectures on Fourier integrals. Princeton, New Jersey: Princeton Univ. Press.

Osypchuk, M. M. & Portenko, M. I. (2016). On single-layer potentials for one class of pseudo-differential equations. Ukr. Math. J., 67, No. 11, pp. 1704-1720. doi: https://doi.org/10.1007/s11253-016-1184-7

Blumenthal, R. M & Getoor, R. K. (1960). Some theorems on stable processes. Trans. Am. Math. Soc., 93, No. 2, pp. 263-273. doi: https://doi.org/10.1090/S0002-9947-1960-0119247-6

Eidelman, S. D., Ivasyshen, S. D. & Kochubei, A. N. (2004). Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Operator Theory: Advances and Applications. (Vol. 152). Basel: Birkhäuser. doi: https://doi.org/10.1007/978-3-0348-7844-9

Osypchuk, M. M. & Portenko, M. I. (2017). On constructing some membranes for a symmetric α-stable process. Commun. Stoch. Anal., 11, No. 1, pp. 11-20. doi: https://doi.org/10.31390/cosa.11.1.02

Bateman, H. & Erdélyi, A. (1953). Higher transcendental functions. (Vol. 2). New York: McGraw-Hill.

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Published

20.05.2024

How to Cite

Osypchuk, M., & Portenko, M. (2024). On the distribution of a rotationally invariant α-stable process at the hitting time of a given hyperplane . Reports of the National Academy of Sciences of Ukraine, (12), 14–20. https://doi.org/10.15407/dopovidi2018.12.014