Arbitrary approximations of first-passage-time distributions in general time-continuous stochastic processes

First-passage time (FPT) is an important quantity characterizing stochastic processes, but its distribution is often difficult to compute. Here we develop an efficient approach to compute the FPT distribution for a general stochastic process modeled by a master equation or a Fokker-Planck equation. The core of this approach is a differential equation for a moment-generating function, which can be solved by expanding this function in some form. As a result, an approximate FPT distribution can be found up to any pregiven accuracy and even the exact FPT distributions can be derived in some special cases. We validate our proposed approach by analyzing several examples and discuss convergence and renormalization issues of the approach. Our approach has broad applications in various disciplines underlying the FPT issues.