Intrinsic small time estimates for distribution densities of L\'evy processes

TL;DR

This paper develops intrinsic small-time estimates for the transition densities of Lévy processes, revealing their structure through characteristic exponents and employing Fourier analysis, with examples including irregular Lévy measures.

Contribution

It introduces a novel method for deriving intrinsic small-time bounds for Lévy process densities based on characteristic exponents and Fourier analysis techniques.

Findings

Established intrinsic upper and lower bounds for Lévy process densities in small time.

Demonstrated the method with examples involving irregular Lévy measures.

Provided insights into the structure of transition densities through characteristic exponents.

Abstract

We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of the process. The technique used in the paper relies on the asymptotic analysis of the inverse Fourier transform of the respective characteristic function. We provide several examples, in particular, with rather irregular L\'evy measure, to illustrate our results.