Explicit asymptotic on first passage times of diffusion processes

We introduce a unified framework for solving first passage times of time- homogeneous diffusion processes. According to the potential theory and the perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for the single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; especially, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein-Uhlenbeck, Bessel, exponential-Shiryaev (studied in [13]), and the hypergeometric diffusion [8] processes. The purpose of this paper is to provide a fast and accurate approach to estimate first passage time densities of various diffusion processes.