Exact Asymptotics for the Probability of Exit from a Domain and Applications to Simulation

Abstract

We study the asymptotics of the exit probability $\mathbb{P}^\varepsilon_{x,s}\{\tau \leq T\}$, where $\tau$ is the exit time from an open set and $\mathbb{P}^\varepsilon_{x,s}$ is the law of a diffusion process with a small parameter $\varepsilon$ multiplying the diffusion coefficient. We consider the case of the Brownian bridge in many dimensions, this choice being motivated by applications to numerical simulation. The method uses recent results reducing the problem to the solution of a system of linear first-order PDE's.