# Asymptotic results for the last zero crossing time of a Brownian motion with non-null drift

## TL;DR

This paper investigates the asymptotic behavior of the last zero crossing time of a Brownian motion with non-zero drift, establishing large deviation principles and comparing them with moderate deviation results.

## Contribution

It provides new large deviation principles for the last zero crossing time of Brownian motion with drift under various scalings, extending previous moderate deviation results.

## Key findings

- Established large deviation principles for $T_{\mu,t}$ as $r \to \infty$
- Derived large deviation principles with different scalings governed by the same rate function
- Compared classical moderate deviation results with new asymptotic findings

## Abstract

We consider the last zero crossing time $T_{\mu,t}$ of a Brownian motion, with drift $\mu \neq 0$ in the time interval $[0, t]$. We prove the large deviation principle of $\{T_{\mu \sqrt r t} : r > 0 \}$ as $r$ tends to infinity. Moreover, motivated by the results on moderate deviations in the literature, we also prove a class of large deviation principles for the same random variables with different scalings, which are governed by the same rate function. Finally we compare some aspects of the classical moderate deviation results, and the results in this paper.