Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

By María Emilia Caballero, L. Chaumont. 2004

66 citations recorded by the source corpus.

Using Lamperti’s relationship between Levy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law ℙx of a pssMp starting at x>0, in the Skorohod space of cadlag paths, when x tends to 0. To do so, we first give conditions which allow us to construct a cadlag Markov process X(0), starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws ℙx to converge weakly to the law of X(0) as x goes to 0. In particular, this answers a question raised by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205–225] about the Feller property for pssMp at x=0.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f5b055c8df0327a08235446bc471f9f84cd1691

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Citations in the corpus

Law of the absorption time of some positive self-similar Markov processes (2009, methodology): “Caballero and Chaumont [11] characterized the characteristic triplet of the underlying Lévy process.”
Some explicit identities associated with positive self-similar Markov processes. (2007, background): “Necessary and sufficient conditions for the weak convergence of (X,Px ) on Skorokhod’s space were given in [9].”
Bernstein-gamma functions and exponential functionals of Levy Processes (2016, background): “The interest in these variables seems to be due to the fact that they appear and play a crucial role in various theoretical and applied contexts such as the spectral theory of some non-reversible Markov semigroups ([61, 60]), the study of random planar maps ([12]), limit theorems of Markov chains ([”