Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

On the lack of memory for distributions of overshoot functionals in the case of upper almost semicontinuous processes defined on a Markov chain


Authors: D. V. Gusak and E. V. Karnaukh
Translated by: S. V. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 99 (2018).
Journal: Theor. Probability and Math. Statist. 99 (2019), 77-89
MSC (2010): Primary 60K37; Secondary 60G51
DOI: https://doi.org/10.1090/tpms/1081
Published electronically: February 27, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the question on whether or not the property of lack of memory that is valid for the geometrical and exponential distributions remains valid for hitting functionals in the case of upper almost semicontinuous processes defined on a Markov chain. If a boundary is attainable and the state of the environment is known at the moment when this boundary is attained, then the lack of memory holds only for an overshoot over a boundary $ x\ge 0$ and the distribution of the overshoot does not depend on the overshoot moment as well as on $ x$. The distribution of an undershoot for a boundary $ x$ is determined via the distribution of the undershoot for a zero boundary. A similar property is proved for a jump crossing a boundary $ x$.


References [Enhancements On Off] (What's this?)

  • [1] S. Asmussen and H. Albrecher, Ruin Probabilities, World Scientific, Singapore, 2010. MR 2766220
  • [2] L. Breuer, From Markov Jump Processes to Spatial Queues, Springer, New York, 2003. MR 1967164
  • [3] A. Pacheco, L. Tang, and N. Prabhu, Markov-Modulated Processes and Semigenerative Phenomena, World Scientific, Singapore, 2008. MR 2484312
  • [4] L. Breuer, First passage times for Markov-additive processes with positive jumps of phase type, J. Appl. Probab. 45 (2008), 779-799. MR 2455184
  • [5] L. Breuer, A quintuple law for Markov additive processes with phase-type jumps, J. Appl. Probab. 47 (2010), 441-458. MR 2668499
  • [6] R. A. Doney and A. E. Kyprianou, Overshoots and undershoots of Lévy processes, Ann. Appl. Probab. 16 (2006), 91-106. MR 2209337
  • [7] L. Breuer and A. L. Badescu, A generalized Gerber-Shiu measure for Markov-additive risk processes with phase-type claims and capital injections, Scand. Actuar. J. 2 (2014), 93-115. MR 3177093
  • [8] A. E. Kyprianou, Gerber-Shiu Risk Theory, Springer, New York, 2013. MR 3135656
  • [9] Z. Jiang and M. Pistorius, On perpetual American put valuation and first-passage in a regime-switching model with jumps, Finance and Stochastics 12 (2008), 331-355. MR 2410841
  • [10] D. V. Husak, Boundary Problems for Processes with Independent Increments on Markov Chains and for Semi-Markov Processes, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1998. (Ukrainian) MR 1710395
  • [11] M. S. Herych and D. V. Husak, On the moment-generating functions of extrema and their complements for almost semicontinuous integer-valued Poisson processes on Markov chains, Ukrain. Matem. Zh. 67 (2016), no. 8, 1034-1049; English transl. in Ukr. Math. J. 67 (2016), no. 8, 1164-1182. MR 3473711
  • [12] D. V. Gusak and E. V. Karnaukh, Matrix factorization identity for almost semi-continuous processes on a Markov chain, Theor. Stoch. Process. 11(27) (2005), No 1-2, 40-47. MR 2327445
  • [13] S. G. Kou and H. Wang, First passage times of a jump diffusion process, Adv. Appl. Prob. 35 (2003), 504-531. MR 1970485
  • [14] D. Husak, Boundary Functionals for Lévy Processes and their Applications, Lambert Academic Publishing, 2014.
  • [15] M. S. Gerych, Distributions of overshoots for almost continuous stochastic processes defined on a Markov chain, Teor. Imovirnost. Matem. Statyst. 94 (2016), 36-49; English transl. in Theor. Probab. Math. Statist. 94 (2017), 37-52. MR 3553452
  • [16] E. V. Karnaukh, Overshoot functionals for almost semicontinuous processes defined on a Markov chain, Teor. Imovirnost. Matem. Statyst. 76 (2007), 45-52; English transl. in Theor. Probab. Math. Statist. 76 (2008), 49-57. MR 2368739
  • [17] D. V. Husak, Boundary Value Problems for Processes with Independent Increments in the Risk Theory, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 2011. (Ukrainian) MR 2382816

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60K37, 60G51

Retrieve articles in all journals with MSC (2010): 60K37, 60G51


Additional Information

D. V. Gusak
Affiliation: Department of Probability Theory and Mathematical Analysis, Faculty of Mathematics, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod, 88000 Ukraine
Email: husakdv@ukr.net

E. V. Karnaukh
Affiliation: Department of Statistics and Probability Theory, Faculty of Mechanics and Mathematics, Oles Honchar Dnipro National University, Gagarin Avenue, 72, Dnipro, 49000 Ukraine
Email: ievgen.karnaukh@gmail.com

DOI: https://doi.org/10.1090/tpms/1081
Keywords: Almost semicontinuous processes defined on a Markov chain, lack of memory, passage functionals for a positive boundary, main factorization identity
Received by editor(s): May 6, 2018
Published electronically: February 27, 2020
Article copyright: © Copyright 2020 American Mathematical Society