Bernstein-gamma functions and exponential functionals of Lévy processes

Abstract

In this work we analyse the solution to the recurrence equationMΨ(z), MΨ(1) = 1, defined on a subset of the imaginary line and where −Ψ runs through the set of all continuous negative definite functions. Using the analytic Wiener-Hopf method we furnish the solution to this equation as a product of functions that extend the classical gamma function. These latter functions, being in bijection with the class of Bernstein functions, are called Bernsteingamma functions. Using their Weierstrass product representation we establish universal Stirling type asymptotic which is explicit in terms of the constituting Bernstein function. This allows the thorough understanding of the decay of |MΨ(z)| along imaginary lines and an access to quantities important for many theoretical and applied studies in probability and analysis.This functional equation appears as a central object in several recent studies ranging from analysis and spectral theory to probability theory. In this paper, as an application of the results above, we investigate from a global perspective the exponential functionals of Lévy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Lévy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations of its successive derivatives. We also furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new Wiener-Hopf and infinite product factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. Some of the results presented in this paper have been announced in the note [60].

A characterization of the finiteness of perpetual integrals of Lévy processes

“…Let f (x) = e −x which is strictly decreasing on R. Then since (−∞, 0) is P-transient and ∞ 0 e −y U (dy) is finite being the U 1 potential, we conclude that the following ∞ 0 e −ξs ds < ∞ holds. This elementary fact is well-known from the theory of exponential functionals of Lévy processes, see Patie, P. and Savov, M. (2018), but illustrates the applicability of our result. We also emphasize that from Lemma 4.6 and the proof of the main theorem for relation (2.4) to hold it suffices to know that the set…”

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

“…In order to avoid the trivial situation when T = ∞ P x -almost surely (a.s.), according to Lamperti, see also [25,Section 2.2], it suffices that…”

“…The sufficient conditions for N = ∞ are easily derived. The fact that ϑ + α (dy) = v + α (y)dy, y > 0, with v + α (0 + ) ∈ (0, ∞) follows from [25,Proposition B.2]. Next, from (2.6), we get that for all t > 0,…”

“…where we used the identities P(λ 1 ≤ s) = P(χ s ≥ 1) = P(χ 1 ≥ s − 1 β ). Then, according to [25,Theorem 2.24], we deduce that for any ℜ(z) > 0,…”

“…where we have used that E x [T cα ] = x α E ∞ 0 exp(αY t )dt cα and the expression of the moments of the latter functional in [25,Theorem 2.4]. Plainly…”