Large Deviations for SDE driven by Heavy-tailed L\'evy Processes

TL;DR

This paper establishes large deviation principles for solutions of 1D SDEs driven by heavy-tailed Lévy processes, highlighting differences from classical results due to the lack of exponential integrability.

Contribution

It extends large deviation theory to SDEs driven by heavy-tailed Lévy processes without exponential integrability, providing new insights into their probabilistic behavior.

Findings

Solutions satisfy a weak large deviation principle with a discrete rate function.

Solutions do not satisfy a full large deviation principle.

The results apply to Lévy processes lacking exponential integrability.

Abstract

We obtain sample-path large deviations for a class of one-dimensional stochastic differential equations with bounded drifts and heavy-tailed L\'evy processes. These heavy-tailed L\'evy processes do not satisfy the exponential integrability condition, which is a common restriction on the L\'evy processes in existing large deviations contents. We further prove that the solution processes satisfy a weak large deviation principle with a discrete rate function and logarithmic speed. We also show that they do not satisfy the full large deviation principle.