Small noise asymptotics for a class of jump-diffusions with heavy tails for large times

TL;DR

This paper studies the long-term behavior of certain jump-diffusion processes with heavy tails under small noise, revealing that their distributions are governed by a combined continuous and impulse control problem.

Contribution

It extends classical small noise asymptotics to jump-diffusions with heavy tails, incorporating both continuous and impulse controls in the analysis.

Findings

Large-time distribution governed by a deterministic control problem

Inclusion of impulse control in the asymptotic analysis

Extension of classical diffusion results to jump processes with heavy tails

Abstract

In this work, we investigate positive recurrent L\'evy diffusions driven by appropriately scaled Brownian motion and $\alpha$-stable process (with $1<\alpha<2$) in the small noise regime. Supposing that in the vanishing noise limit, our L\'evy diffusion approaches a deterministic system with a unique asymptotically stable fixed point, we show that the limiting behavior of the one-dimensional marginal distribution at large times is dictated by the optimal value of a deterministic control problem, just as in the classical case of diffusions driven by small variance Brownian motion. In our case, the control is allowed to have two parts: continuous control and impulse control.