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Transactions of the American Mathematical Society
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
Published electronically: August 18, 2009
MathSciNet review: 2550159
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-09-04791-6
PII: S 0002-9947(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.