Fluctuations of Lévy processes and scattering theory

Fourati, Sonia

doi:10.1090/S0002-9947-09-04791-6

Fluctuations of Lévy processes and scattering theory
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Trans. Amer. Math. Soc. 362 (2010), 441-475 Request permission

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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