Exponential functionals of Lévy and Markov additive Lévy processes

Victor Rivero (Centro de Investigacion en Matematicas)

Williamson Building, G.33,

Let I be the exponential functional of a real valued Lévy process  This random variable plays a key rol in several areas of probability theory, for instance in fragmentation, coalescence and branching processes, financial and insurance mathematics, Brownian motion in hyperbolic spaces, random processes in random environment, positive self-similar Markov processes, etc. Obtaining distributional properties of these r.v. has been the subject of many research articles. The paper by Bertoin and Yor [1] is a thorough review on the topic. In this talk we will show that the law of this random variable satisfies an functional equation in terms of the potential measure of the underlying Levy process. We will show how this identity can be used to provide elementary proofs of several known results for I, in particular of the striking factorisation obtained in [2]. To finish we will establish that a similar identity holds when we replace ξ by a Markov additive Lévy process, and explain how our methods allow to derive analogous results to the Lévy processes case, in particular of the main result in [2].

 

References

[1] J. Bertoin and M. Yor. Exponential functionals of L ́evy processes. Probab. Surv., 2:191–212, 2005.

[2] P. Patie and M. Savov. Exponential functional of L ́evy processes: generalized Weierstrass products and Wiener-Hopf factorization. C. R. Math. Acad. Sci. Paris, 351(9-10):393–396, 2013.

 

 

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