On extrema of stable processes.

By Alexey Kuznetsov. 2010

62 citations recorded by the source corpus.

We study the Wiener--Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener--Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener--Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/727eeff0969b3bc72872caac00f61209db80d820

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes (2010, methodology): “Barnes has shown that it is possible to define constants a = a(τ) and b = b(τ) in such a way so that we have G(1; τ) = 1 and that G(z; τ) satisfies the following three functional identities (see [2] and [27])”
Some Properties of the Kilbas-Saigo Function (2020, methodology): “tenation formula G(z+δ;δ) = (2π)(δ−1)/2δ1/2−zΓ(z)G(z;δ) (A.5) which is valid for all z∈ C,the right-hand side being understood as an analytic extension when z is a non-positive integer - see (4.6) in [15] and the references therein. Observe that (A.1) and (A.5) lead readily to the closed formula G(δ”
Suprema of Lévy processes (2011, background): “There are numerous other representations for the distribution of Mt, at least in the stable case, see [4, 7, 11, 12, 15, 16, 19, 20, 27, 28, 30, 36].”
Bernstein-gamma functions and exponential functionals of Levy Processes (2016, methodology): “The latter for example immediately recovers the asymptotic behaviour of the density of the supremum of a stable Lévy process as investigated in [8, 25, 40, 56].”