Bernstein-gamma functions and exponential functionals of Levy Processes

We study the equation $M_\Psi(z+1)=\frac{-z}{\Psi(-z)}M_\Psi(z), M_\Psi(1)=1$ defined on a subset of the imaginary line and where $\Psi$ is a negative definite functions. Using the Wiener-Hopf method we solve this equation in a two terms product which consists of functions that extend the classical gamma function. These functions are in a bijection with Bernstein functions and for this reason we call them Bernstein-gamma functions. Via a couple of computable parameters we characterize of these functions as meromorphic functions on a complex strip. We also establish explicit and universal Stirling type asymptotic in terms of the constituting Bernstein function. The decay of $|M_{\Psi}(z)|$ along imaginary lines is computed. Important quantities for theoretical and applied studies are rendered accessible. As an application we investigate the exponential functionals of Levy Processes whose Mellin transform satisfies the recurrent equation above. Although these variables have been intensively studied, our new perspective, based on a combination of probabilistic and complex analytical techniques, enables us to derive comprehensive and substantial properties and strengthen several results on the law of these random variables. These include smoothness, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations for the density and its successive derivatives. We also study the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. The derivation of our results relies on a mixture of complex-analytical and probabilistic techniques.

1 W φ appears crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. The quantification of its analytic properties offers explicit information about eigen-and coeigen-functions, their norms, etc.
2 W φ are related to the "phenomenon of self-similarity" the same way the Gamma function appears in some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ appears in exponential functionals of Lévy processes 1 W φ appears crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. The quantification of its analytic properties offers explicit information about eigen-and coeigen-functions, their norms, etc.
2 W φ are related to the "phenomenon of self-similarity" the same way the Gamma function appears in some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ appears in exponential functionals of Lévy processes 1 W φ appears crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. The quantification of its analytic properties offers explicit information about eigen-and coeigen-functions, their norms, etc.
2 W φ are related to the "phenomenon of self-similarity" the same way the Gamma function appears in some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ appears in exponential functionals of Lévy processes Denote by the set of all Lévy-Khintchine exponents of possibly killed at exponential random time of parameter q ≥ 0 Lévy processes.

Denote by
the set of all Lévy-Khintchine exponents of possibly killed at exponential random time of parameter q ≥ 0 Lévy processes.
The random variables I Ψ = eq 0 e −ξs ds, e q ∼ Exp(q); e 0 = ∞ are called exponential functionals of Lévy processes and For any Ψ ∈ N to solve and characterize the solutions of in terms of the global quantities of Ψ 2 Use that M I Ψ (z + 1) = E [I z Ψ ] solves in some sense (0.2) to obtain information about I Ψ 1 For any Ψ ∈ N to solve and characterize the solutions of in terms of the global quantities of Ψ 2 Use that M I Ψ (z + 1) = E [I z Ψ ] solves in some sense (0.2) to obtain information about I Ψ 1 Appear in financial and insurance mathematics; branching with immigration; fragmentation; self-similar growth fragmentation; etc. 2 We also use it in our work on the spectral theory of positive self-similar semigroups 1 Appear in financial and insurance mathematics; branching with immigration; fragmentation; self-similar growth fragmentation; etc. 2 We also use it in our work on the spectral theory of positive self-similar semigroups 1 I Ψ introduced and studied by Urbanik when ξ is a subordinator 2 Further studied by Carmona, Petit and Yor who have in special cases Further studied by Carmona, Petit and Yor who have in special cases Maulik and Zwart derive this key recurrent equation in some generality and utilize it 4 Kuznetsov solves this recurrent relation for some classes of Lévy processes 5 There are various other contributions relying on different approaches We use A (a,b) (resp. M (a,b) ) to denote the holomorphic (resp. meromorphic) functions on the complex strip We use A (a,b) (resp. M (a,b) ) to denote the holomorphic (resp. meromorphic) functions on the complex strip is a solution to f(z + 1) = φ(z)f(z), f(1) = 1. Theorem is a solution to f(z + 1) = φ(z)f(z), f(1) = 1. Moreover, W φ is zero-free on is a solution to f(z + 1) = φ(z)f(z), f(1) = 1. Moreover, W φ is zero-free on

M. Savov and P. Patie
Bernstein-gamma functions and exponential functionals For B P = {φ ∈ B : δ > 0} then the asymptotic along a + iR is For B P = {φ ∈ B : δ > 0} then the asymptotic along a + iR is For B P = {φ ∈ B : δ > 0} then the asymptotic along a + iR is The celebrated factorization Wiener-Hopf are Bernstein functions then yields The product of the solutions to the independent system on a common complex domain is a solution to f (z + 1) The product of the solutions to the independent system on a common complex domain is a solution to f (z + 1) = −z Ψ(−z) f(z) on this domain.
These can be extracted from the general solution to f ± (z + 1) As a consequence of the Weierstrass product representations of W φ± , Γ
N Ψ is a measure for the polynomial decay of |M Ψ | along complex lines.