Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fluctuations of Lévy processes and scattering theory


Author: Sonia Fourati
Journal: Trans. Amer. Math. Soc. 362 (2010), 441-475
MSC (2000): Primary 60G51, 34L25; Secondary 60G52, 35Q15
DOI: https://doi.org/10.1090/S0002-9947-09-04791-6
Published electronically: August 18, 2009
MathSciNet review: 2550159
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Initial work by Spitzer was extended to show that the behavior of the bivariate processes $ \displaystyle(X_t,\inf_{0\leq s\leq t}X_s)$ or $ \displaystyle (X_t,\sup _{0\leq s\leq t}X_s)$, where $ X$ is a Lévy process, can be entirely reconstructed on the basis of the Wiener-Hopf factorization of the Lévy exponent of $ X$. This paper is meant to establish that a similar device can be used to investigate the trivariate Markov process $ \displaystyle (X_t,\inf_{0\leq s\leq t}X_s,\sup_{0\leq s\leq t} X_s)$. This involves substituting (2,2)-matrices for the scalar functions involved in the Spitzer-type factorization. The computation of this matrix from the Lévy exponent of $ X$ is a Riemann-Hilbert problem, which is the same as the one appearing in the inverse scattering problem.


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

  • 1. Avram, F., Kyprianou, A.E., Pistorius, M.R. Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14, 215-238 (2004). MR 2023021 (2005c:60053)
  • 2. Bertoin, J. Exponential decay and geometric ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7, 156-167 (1997). MR 1428754 (98d:60147)
  • 3. Bertoin, J. Lévy processes. Cambridge University Press (1996). MR 1406564 (98e:60117)
  • 4. Bingham, N.H. Fluctuation theory in continuous time. Adv. Appl. Probab. 7, 705-766, (1975). MR 0386027 (52:6886)
  • 5. Beals, R., Deift, P., Zhou, X. The inverse scattering transform on the line. Important Developments in Soliton Theory. A.S. Fokas, V.E. Zakharov (Eds). Springer-Verlag 7-33.(1993) MR 1280467 (95k:34020)
  • 6. Doney, R. A. Some excursion calculations for spectrally one-sided Lévy processes. Sem. de Probab. XXXVIII, Lecture Notes in Math. 1857, 5-15, (2004). MR 2126963 (2006b:60098)
  • 7. Emery, D. J. Exit problem for a spectrally positive process. Adv. Appl. Probab. 5, 498-520, (1973). MR 0341623 (49:6370)
  • 8. Fourati, S. Points de croissance des processus de Lévy et théorie générale des processus. Probab.Theory Relat. Fields 110 13-49, (1998). MR 1602032 (99e:60164)
  • 9. Fourati, S. Krein theory on strings applied to fluctuations of Lévy processes. Preprint. ArXiv: Math.PR/0508612v1 (2005).
  • 10. Fourati, S. Fluctuation des processus de Lévy et dispersion (``scattering'').C.R. Acad. Sci. 1342, 135-139, (2006). MR 2193661
  • 11. Karandov,V.F. and Karandova, T.V. On the distribution of the time of the first exit from an interval and value of a jump over the boundary for processes with independent increments and random walks. Ukainian Mathematical Journal Vol. 47, No. 10 (2005).
  • 12. Koryluk, V.S., Suprun, V.N. and Shurenkov, V. M. Method of potential in boundary problems for processes with increases and jumps of the same sign. Theory. Probab. Appl. 21, 243-249(1976).
  • 13. Kyprianou, A. E. First passage of reflected strictly stable processes ALEA 2, 119-123, (2006). MR 2249665 (2008e:60139)
  • 14. Kyprianou, A. E. and Palmowski, Z. A martingale review of some fluctuation theory for spectrally negative Lévy processes. Sem. de Probab. XXXVIII, Lecture Notes in Math. 1857, 16-29, (2004). MR 2126964 (2005m:60103)
  • 15. Nguyen-Ngoc, L. and Yor, M. Some martingales associated to reflected Lévy processes. Sem. de Probab. XXXVIII, Lecture Notes in Math., no. 1857, 42-69, (2005). MR 2126966 (2006a:60081)
  • 16. Pistorius, M. R. On exit and ergodicity of the spectrally negative Lévy process at its infimum. J. Theor. Probab. 17, 183-220, (2004). MR 2054585 (2005e:60104)
  • 17. Pistorius, M. R. A potential theoretical review of some exit problems of spectrally negative Lévy processes. Sem. de Probab. XXXVIII, Lecture Notes in Math 1857, 42-69, (2004). MR 2126965 (2006k:60086)
  • 18. Rogers, L. C. G. The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Probab. 22, 486-487, (1990). MR 1053243 (93f:60110)
  • 19. Rogozin, B. A. The distribution of the position of absorption for stable and asymptotically stable random walks on an interval. Theor. Probab. Appl. 17 (1972), 332-338. MR 0300349 (45:9395)
  • 20. Sato, K. Lévy processes and infinitely divisible distributions. Cambridge University Press (1999). MR 1739520 (2003b:60064)
  • 21. Shabat, A. B. An inverse scattering problem. Differential Equations 15 1299-1307, (1980). MR 553630 (81m:34026)
  • 22. Spitzer, F. A combinatorial lemma and its applications to probability theory.Trans. Amer. Math. Soc. 82 323-339, (1956). MR 0079851 (18:156e)
  • 23. Suprun, V. N. The ruin problem and the resolvent of a killed independent increment process. Ukrainian Math. J. 28, 39-45, (1976). MR 0428476 (55:1497)
  • 24. Takács, L. Combinatorial methods in the theory of stochastic processes. Wiley, New York (1967. ). MR 0217858 (36:947)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60G51, 34L25, 60G52, 35Q15

Retrieve articles in all journals with MSC (2000): 60G51, 34L25, 60G52, 35Q15


Additional Information

Sonia Fourati
Affiliation: Laboratoire de Probabilities, University of Paris VI, 4 Place Jussieu Tour 56, 75252 Paris Cedex 5, France
Address at time of publication: Place Emile Blondel 76131 Mont Saint Aignan, France
Email: sonia.fourati@upmc.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04791-6
Keywords: L\'evy processes, fluctuation theory, Wiener-Hopf factorization, scattering theory, Riemann-Hilbert factorization
Received by editor(s): February 8, 2007
Received by editor(s) in revised form: March 28, 2008
Published electronically: August 18, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society