Electronic Journal of Probability

Bernstein-gamma functions and exponential functionals of Lévy processes

Pierre Patie and Mladen Savov

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In this work we analyse the solution to the recurrence equation \[ M_\Psi (z+1)=\frac{-z} {\Psi (-z)}M_\Psi (z), \quad M_\Psi (1)=1, \] defined on a subset of the imaginary line and where $-\Psi $ is any continuous negative definite function. Using the analytic Wiener-Hopf method we solve this equation as a product of functions that extend the gamma function and are in bijection with the Bernstein functions. We call these functions Bernstein-gamma functions. We establish universal Stirling type asymptotic in terms of the constituting Bernstein function. This allows the full understanding of the decay of $\vert M_\Psi (z)\vert $ along imaginary lines and an access to quantities important for many studies in probability and analysis.

This functional equation is a central object in several recent studies ranging from analysis and spectral theory to probability theory. As an application of the results above, we study 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 and bounds. We furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. We deliver intertwining relation between members of the class of positive self-similar semigroups.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 75, 101 pp.

Received: 8 October 2016
Accepted: 20 July 2018
First available in Project Euclid: 27 July 2018

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Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 60G51: Processes with independent increments; Lévy processes 60J55: Local time and additive functionals 60E07: Infinitely divisible distributions; stable distributions 44A60: Moment problems 33E99: None of the above, but in this section

asymptotic analysis functional equations exponential functional Lévy processes Wiener-Hopf factorizations special functions intertwining Bernstein functions

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Patie, Pierre; Savov, Mladen. Bernstein-gamma functions and exponential functionals of Lévy processes. Electron. J. Probab. 23 (2018), paper no. 75, 101 pp. doi:10.1214/18-EJP202. https://projecteuclid.org/euclid.ejp/1532678638

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