Exact asymptotic for distribution densities of Levy functionals

TL;DR

This paper develops a saddle point method to precisely describe the asymptotic behavior of distribution densities for Levy-driven stochastic integrals, including Levy processes, Ornstein-Uhlenbeck processes, and fractional Levy motion.

Contribution

It introduces an exact asymptotic analysis technique for Levy functionals, extending understanding of their distribution densities in various stochastic models.

Findings

Exact asymptotics for Levy process densities

Asymptotic behavior of Levy-driven Ornstein-Uhlenbeck process densities

Distribution density of fractional Levy motion analyzed

Abstract

A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a) the transition probability density of a real-valued Levy process; (b) the transition probability density and the invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c) the distribution density of the fractional Levy motion.