Large deviations for the boundary local time of doubly reflected Brownian Motion

TL;DR

This paper derives a closed-form expression for the moment generating function of the boundary local time of doubly reflected Brownian motion, and uses it to establish a large deviation principle for the local time growth rate.

Contribution

It provides a novel explicit formula for the moment generating function and applies large deviation theory to analyze the boundary local time of doubly reflected Brownian motion.

Findings

Explicit formula for the moment generating function of boundary local time.

Large deviation principle for the local time growth rate.

Analysis of the blow-up behavior of the generating function.

Abstract

We compute a closed-form expression for the moment generating function $\hat{f}(x;\lambda,\alpha)=\frac{1}{\lambda}\mathbb{E}_x(e^{\alpha L_{\tau}})$, where $L_t$ is the local time at zero for standard Brownian motion with reflecting barriers at $0$ and $b$, and $\tau \sim \mathrm{Exp}(\lambda)$ is independent of $W$. By analyzing how and where $\hat{f}(x;\cdot,\alpha)$ blows up in $\lambda$, a large-time large deviation principle (LDP) for $L_t/t$ is established using a Tauberian result and the G\"{a}rtner-Ellis Theorem.