Exponential functionals of Levy processes

By Jean Bertoin, Marc Yor. 2005

279 citations recorded by the source corpus.

The distribution of the terminal value A∞ of the exponential functional $$ {A_t}(\xi ) = \smallint _0^t{e^{{\xi _s}}}ds $$ of a Levy process (ξ t ) t≥0 plays an important role in Mathematical Physics and Mathematical Finance. We show how this distribution can be computed by means of Lamperti’s transformation and generalized Ornstein-Uhlenbeck processes.

Subfields: Control and Systems Engineering, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40e94eb56cb2b51394b49316cc17562ee0e7d7d9

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In Control and Systems Engineering

In Finance

In Mathematical Physics

Citations in the corpus

Bernstein-gamma functions and exponential functionals of Levy Processes (2016, background): “([18]), random affine equations, perpetuities, etc.”
Integrated insurance risk models with exponential Lévy investment (2008, background): “The following examples can be found in Carmona, Petit and Yor [1].”
Random recurrence equations and ruin in a Markov-dependent stochastic economic environment (2009, result): “Nonetheless, (D1) could also be viewed as a possible starting point for our results, and this approach has recently been followed, for example, by Chan and Lai (2007) and Balaji and Meyn (2000).”
Exponential functionals of Brownian motion, I: Probability laws at fixed time (2005, background): “As is mentioned above, the exponential perpetual functionals of Lévy processes have been studied in [6] and [7].”
Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims (2009, background): “Lindner and Maller (2005) or Carmona et al. (2001). Denote the Laplace exponents of the processes L and Lθ by ψ(s) = log E[e−sL(1)] and ψθ(s) = log E[exp(−sLθ (1))], provided that they exist.”
On the extinction of Continuous State Branching Processes with catastrophes (2013, background): “The asymptotic behavior of exponential functionals of L´evy processes drifting to + has been deeply studied by many authors, see for instance Bertoin and Yor [BY05] and references therein.”
Asymptotic results for exponential functionals of Levy processes (2016, background): “The proofs for regimes (4) and (5) are also based on transformations of the underlying Lévy process.”
Law of the absorption time of some positive self-similar Markov processes (2009, background): “We mention the papers [14], [19] and [35] where such examples can be found and refer to the survey paper [8] for a description of these cases.”
Recurrent extensions of self-similar Markov processes and Cramér’s condition II (2005, background): “Jo A direct consequence of Theorem 1 of [4] and the fact that (?, P&) is a Levy process that drifts to oo is that J < oo, P^-a.”
Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process (2007, methodology): “See, for example, the survey in Bertoin and Yor (2005). Not surprisingly, some of our calculations are quite close in nature to those presented in the latter article.”
Conditioned stable Levy processes and Lamperti representation. (2006, background): “The process (X,Px) is a pssMp which belongs to the class C2, with index α = 2 and it is well known (see for instance [6]) that its associated Lévy process in the Lamperti representation (1.”
Some explicit identities associated with positive self-similar Markov processes. (2007, background): “See Bertoin and Yor [6] for an overview on this topic.”