Sample-Path Large Deviations for L\'evy Processes and Random Walks with Lognormal Increments

TL;DR

This paper develops an extended large deviation principle for Le9vy processes and random walks with lognormal increments, filling a gap in the understanding of heavy-tailed processes in the lognormal case.

Contribution

It establishes the first sample-path level extended LDP for lognormal-type increments, including multi-dimensional cases, and demonstrates the limits of these results.

Findings

Extended LDPs proven for one-dimensional processes

Extended LDPs proven for multi-dimensional processes with independent coordinates

Counterexamples show standard LDPs cannot be achieved under certain topologies

Abstract

The large deviations theory for heavy-tailed processes has seen significant advances in the recent past. In particular, Rhee et al. (2019) and Bazhba et al. (2020) established large deviation asymptotics at the sample-path level for L\'evy processes and random walks with regularly varying and (heavy-tailed) Weibull-type increments. This leaves the lognormal case -- one of the three most prominent classes of heavy-tailed distributions, alongside regular variation and Weibull -- open. This article establishes the \emph{extended large deviation principle} (extended LDP) at the sample-path level for one-dimensional L\'evy processes and random walks with lognormal-type increments. Building on these results, we also establish the extended LDPs for multi-dimensional processes with independent coordinates. We demonstrate the sharpness of these results by constructing counterexamples, thereby proving that our results cannot be strengthened to a standard LDP under $J_1$ topology and $M_1'$ topology.