A Note on Wiener–Hopf Factorization for Markov Additive Processes

详细信息    查看全文

• 作者：Przemys?aw Klusik (1)
  Zbigniew Palmowski (1)
  
• 关键词：Levy process ; Wiener–Hopf factorization ; 60G05 ; 60G51 ; 60J25 ; 60J75
• 刊名：Journal of Theoretical Probability
• 出版年：2014
• 出版时间：March 2014
• 年：2014
• 卷：27
• 期：1
• 页码：202-219
• 全文大小：615 KB
• 参考文献：1. Asmussen, S.: Stationary distributions for fluid models with or without Brownian noise. Stoch. Models 11, 21-9 (1995) CrossRef
  2. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
  3. Asmussen, S., Kella, O.: A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab. 32(2), 376-93 (2000) CrossRef
  4. Asmussen, S., Avram, F., Pistorius, M.: Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79-11 (2004) CrossRef
  5. Arjas, E., Speed, T.P.: Symmetric Wiener–Hopf factorization in Markov additive processes. Z.?Wahrscheinlichkeitstheor. Verw. Geb. 26, 105-18 (1973) CrossRef
  6. D’Auria, B., Ivanovs, J., Kella, O., Mandjes, M.: First Passage Process of a Markov additive process and generalized Jordan chains. J. Appl. Probab. 47(4), 1048-057 (2010) CrossRef
  7. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)
  8. Bertoin, J.: Subordinators, Lévy Processes with no Negative Jumps, and Branching Processes. Notes from the course given at MaPhySto, Denmark (2000)
  9. Breuer, L.: First passage times for Markov-additive processes with positive jumps of phase-type. J.?Appl. Probab. 45(3), 779-99 (2008) CrossRef
  10. Breuer: Exit problems for reflected Markov-additive processes with phase-type jumps. J. Appl. Probab. 49(3) (2012)
  11. Conway, J.B.: Functions of one Complex Variable, 2nd edn. Springer, Berlin (1978) CrossRef
  12. Dieker, T., Mandjes, M.: Extremes of Markov-additive processes with one-sided jumps, with queueing applications. Methodol. Comput. Appl. Probab. 13, 221-67 (2011) CrossRef
  13. Feller, W.: An Introduction to Probability Theory and Its Applications, vol.?II, 2nd edn. Wiley, New York (1970)
  14. Fourati, S.: Fluctuations of Lévy processes and scattering theory. Trans. Am. Math. Soc. 362(1), 441-47 (2010) CrossRef
  15. Fourati, S.: Explicit solutions of the exit problem for a class of Lévy processes. Applications to the pricing of double barrier options. Stoch. Proc. Appl. 122(3), 1034-067 (2012) CrossRef
  16. Fristedt, B.E.: Sample functions of stochastic processes with stationary independent increments. Adv. Probab. Relat. Top. 3, 241-96 (1974)
  17. Greenwood, P., Pitman, J.W.: Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12, 893-02 (1980) CrossRef
  18. Ivanovs, J., Boxma, O., Mandjes, M.: Singularities of the generator of a Markov additive process with one-sided jumps. Stoch. Process. Appl. 120(9), 1776-794 (2010) CrossRef
  19. Kaspi, H.: On the symmetric Wiener–Hopf factorization for Markov additive processes. Z.?Wahrscheinlichkeitstheor. Verw. Geb. 59, 179-96 (1982) CrossRef
  20. Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin (2006)
  21. Kyprianou, A.E., Palmowski, Z.: Fluctuations of spectrally negative Markov additive processes. Sém. Probab. XLI, 121-35 (2008)
  22. Maissoneuve, B.: Exit systems. Ann. Probab. 3, 399-11 (1975) CrossRef
  23. Palmowski, Z., Rolski, T.: A technique for the exponential change of measure for Markov processes. Bernoulli 8(6), 767-85 (2002)
  24. Pistorius, M.R.: A potential theoretical review of some exit problems of spectrally negative Lévy processes. Sém. Probab. 38, 30-1 (2005)
  25. Prabhu, N.U.: Stochastic Storage Processes. Queues, Insurance Risk, Dams, and Data Communication, 2nd edn. Springer, New York (1998) CrossRef
  26. Presman, E.L.: Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. Izv. Akad. Nauk USSR 3, 815-52 (1969) CrossRef
  27. Rogers, L.C.G.: Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Probab. 4, 390-13 (1994) CrossRef
  28. Rogozin, B.A.: On the distribution of functionals related to boundary problems for processes with independent increments. Theory Probab. Appl. 11, 580-91 (1966) CrossRef
  29. Spitzer, F.: Principles of Random Walk. Van Nostrand, New York (1964) CrossRef
  
• 作者单位：Przemys?aw Klusik (1)
  Zbigniew Palmowski (1)
  
  1. University of Wroc?aw, pl. Grunwaldzki 2/4, 50-384, Wroc?aw, Poland
  
• ISSN：1572-9230

文摘

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.