[Paper Review] Asymptotic of densities of exponential functionals of subordinators

Martin Minchev, Mladen Savov|arXiv (Cornell University)|Apr 12, 2021

Stochastic processes and statistical mechanics35 references4 citations

TL;DR

This paper establishes non-classical Tauberian asymptotics for the tail, density, and derivatives of exponential functionals of subordinators under a positive increase condition on the Lévy measure. By refining the saddle point method applied to Mellin transforms expressed via Bernstein-gamma functions, it derives precise asymptotic behavior—showing that densities of exponential functionals of non-decreasing compound Poisson processes match those of exponential distributions, and that many such densities are analytic in complex cones.

ABSTRACT

In this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the Levy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein-gamma functions. To apply the saddle point method we improved the Stirling type of asymptotic for Bernstein-gamma functions and the latter is of interest beyond this paper as the Bernstein-gamma functions are applicable in different settings especially through their asymptotic behaviour in the complex plane. As an application we have derived the asymptotic of the density and its derivatives for all exponential functionals of non-decreasing, potentially compound Poisson processes which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.

Motivation & Objective

• To derive asymptotic expansions for the tail, density, and derivatives of exponential functionals of subordinators.
• To establish non-classical Tauberian theorems at infinity under the positive increase condition on the Lévy measure.
• To extend the applicability of the saddle point method to Mellin transforms involving Bernstein-gamma functions.
• To characterize the analyticity properties of such densities in the complex plane.
• To demonstrate that exponential functionals of non-decreasing compound Poisson processes exhibit exponential-type tail decay.

Proposed method

• Application of the saddle point method to the Mellin transform of exponential functionals, which are expressed in terms of Bernstein-gamma functions.
• Development of a refined Stirling-type asymptotic for Bernstein-gamma functions in the complex plane.
• Use of the positive increase condition on the Lévy measure to ensure regularity in asymptotic behavior.
• Analytic continuation and contour deformation techniques to evaluate inverse Mellin transforms.
• Exploitation of the structure of subordinators and their Laplace exponents to derive density asymptotics.
• Verification of analyticity of densities in a sector of the complex plane via uniform bounds on the saddle point expansion.

Experimental results

Research questions

• RQ1 How do the tails and densities of exponential functionals of subordinators behave asymptotically at infinity under mild Lévy measure conditions?
• RQ2 To what extent can the saddle point method be adapted to Mellin transforms involving Bernstein-gamma functions?
• RQ3 What is the precise asymptotic form of the density and its derivatives for exponential functionals of non-decreasing compound Poisson processes?
• RQ4 In which regions of the complex plane are the densities of such functionals analytic?
• RQ5 Can the asymptotic behavior of these densities be characterized as equivalent to that of an exponential distribution?

Key findings

• The density and its derivatives of exponential functionals of non-decreasing compound Poisson processes exhibit asymptotic behavior identical to that of an exponentially distributed random variable.
• The asymptotic expansion of the Mellin transform via Bernstein-gamma functions is extended to a sharp Stirling-type approximation in the complex plane.
• The saddle point method yields precise asymptotic expansions for the tail and density, even for higher-order derivatives.
• A large class of densities arising from exponential functionals are analytic in a sector (cone) of the complex plane, indicating strong regularity.
• The positive increase condition on the Lévy measure is sufficient to ensure the validity of the derived asymptotics across the entire class of considered subordinators.
• The results extend beyond classical Tauberian theory, providing non-classical asymptotic results for a broad class of self-decomposable distributions.