Numerical solution of first passage problems using an approximate Chapman-Kolmogorov relation

A new deterministic numerical method for solving first passage time problem is described, analyzed and computationally tested. The method is based on recursively solving an integral equation for the reliability function. The integral equation is derived from the Chapman-Kolmogorov relation and involves an approximation to the Greenʹs function for the forward Kolmogorov equation. An error analysis yields estimates of convergence rates. Numerical experiments indicate that the method is stable and can accurately approximate the reliability function and first passage times.