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

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