Conditioned stable Levy processes and Lamperti representation.

By María Emilia Caballero, L. Chaumont. 2006

94 citations recorded by the source corpus.

By killing a stable Levy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti (10). For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the char- acteristics of the underlying Levy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Levy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponen- tial time of a certain class of Levy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Levy processes.

Subfields: Finance, Management Information Systems, Management Science and Operations Research

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Citations in the corpus

Old and New Examples of Scale Functions for Spectrally Negative Levy Processes (2007, background): “See also [16] for the origin of the underling Lévy processes.”
The Lamperti representation of real-valued self-similar Markov processes (2011, methodology): “Caballero and Chaumont in [3] proved that the Lévy process ξ related to X via Lamperti’s representation has the characteristic exponent:”
Law of the absorption time of some positive self-similar Markov processes (2009, background): “Caballero and Chaumont [11] characterized the characteristic triplet of the underlying Lévy process.”
Meromorphic Lévy processes and their fluctuation identities. (2010, background): “We mention, in particular, the many cases of spectrally negative Lévy processes in [20, 33], hyper-exponential and generalized hyper-exponential Lévy processes [23], Lamperti-stable processes in [9, 10, 13], Hypergeometric processes in [30, 32], β-processes in [28] and θ -processes in [29].”
Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes (2010, methodology): “The three Lamperti-stable processes ξ∗, ξ↑ and ξ↓ were introduced by Caballero and Chaumont [10] by applying the Lamperti transformation (see [34]) to the positive self-similar Markov processes construced from a stable process.”
A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “Here one also finds a growing body of work (e.g.: Doney and Kyprianou, 2006; Klüppelberg and Erder, 2008; Breuer, 2008; Kyprianou, Pardo and Rivero, 2008a; Chaumont, Kyprianou and Pardo, 2009; Caballero and Chaumont, 2006; Chen and Sheu, 2009).”
Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes (2010, background): “This result is very similar to Proposition 1 in [11], where the authors are able to compute the law of Iτ(q) in closed form only for a single value of q, and this law is essentially identical to the distribution of Sτ(4).”
Some explicit identities associated with positive self-similar Markov processes. (2007, methodology): “The characteristic exponent of ξ has been computed in [10] and is given by”