On the distribution of functionals of the subordinator
Author:
D. V. Gusak
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 88 (2014), 51-66
MSC (2010):
Primary 60G50; Secondary 60K10
DOI:
https://doi.org/10.1090/S0094-9000-2014-00918-7
Published electronically:
July 24, 2014
MathSciNet review:
3112634
Full-text PDF Free Access
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Abstract: The distributions of boundary functionals have been studied by many authors for stochastic processes with independent stationary increments. (See, for example, [1]–[6].) If a process possesses a bounded variation and the drift is nonnegative, then one can obtain (see [4, 5]) the relations determining the distributions of some functionals of the subordinator defined as a stochastic process with independent positive increments; see Chapter III of [3].
References
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- A. A. Borovkov, Stochastic processes in queueing theory, Springer-Verlag, New York-Berlin, 1976. Translated from the Russian by Kenneth Wickwire; Applications of Mathematics, No. 4. MR 0391297
- Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
- D. V. Gusak, Distribution of overshoot functionals of a semicontinuous homogeneous process with independent increments, Ukraïn. Mat. Zh. 54 (2002), no. 3, 303–322 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 3, 371–397. MR 1952790, DOI https://doi.org/10.1023/A%3A1020557215381
- D. V. Gusak, Processes with Independent Increments in the Risk Theory, Proceedings of the Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 2011. (Ukrainian)
- N. S. Bratiĭchuk and D. V. Gusak, Granichnye zadachi dlya protsessov s nezavisimymi prirashcheniyami, “Naukova Dumka”, Kiev, 1990 (Russian). With an English summary. MR 1070711
- D. V. Gusak, A. G. Kukush, A. M. Kulik, Yu. S. Michura, and A. Yu. Pilipenko, A Collection of Problems in the Theory of Stochastic Processes and their Applications, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, and Andrey Pilipenko, Theory of stochastic processes, Problem Books in Mathematics, Springer, New York, 2010. With applications to financial mathematics and risk theory. MR 2572942
References
- V. S. Korolyuk, Boundary Problems for Compound Poisson Processes, “Naukova dumka”, Kiev, 1975. (Russian) MR 0402939 (53:6753)
- A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer-Verlag, Berlin, 1976. MR 0391297 (52:12118)
- J. Bertoin, Lévy Processes, Cambrige University Press, Cambrige, 1996. MR 1406564 (98e:60117)
- D. V. Gusak, Distribution of overjump functionals of a semicontinuous homogeneous process with independent increments, Ukr. Mat. Zh. 54 (2002), no. 3, 303–322; English transl. in Ukrainian Math. J. 54 (2002), no. 3, 371–397. MR 1952790 (2003j:60067)
- D. V. Gusak, Processes with Independent Increments in the Risk Theory, Proceedings of the Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 2011. (Ukrainian)
- N. S. Bratiĭchuk and D. V. Gusak, Boundary Problems for Processes with Independent Increments, “Naukova dumka”, Kiev, 1990. (Russian) MR 1070711 (91m:60139)
- D. V. Gusak, A. G. Kukush, A. M. Kulik, Yu. S. Michura, and A. Yu. Pilipenko, A Collection of Problems in the Theory of Stochastic Processes and their Applications, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- D. Gusak, A. Kukush, A. Kulik, Yu. Michura, and A. Pilipenko, Theory of Stochastic Processes. With applications to Financial Mathematics and Risk Theory, Springer, New York–Dordrecht–Heidelberg–London, 2010. MR 2572942 (2011f:60069)
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Additional Information
D. V. Gusak
Affiliation:
252601, Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street 3, Kyiv, Ukraine
Email:
random@imath.kiev.ua
Keywords:
Stochastic processes with stationary independent increments,
subordinator,
cumulant and potential of stochastic processes
Received by editor(s):
December 13, 2011
Published electronically:
July 24, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
