A deterministsic numerical method for solutions of first-passage time problems
In this research, a new deterministic numerical procedure for first passage time problems is introduced, analyzed, and numerically tested. In this procedure, the Green's function solution to the forward Kolmogovorov equation is approximated for a small time step. The reliability function is then approximated by recursively and numerically solving an iterated integral whose integrand involves the approximate Green's function. The reliability function, which solves the backward Kolmogorov equation, yields the probability distribution of first passage times, and hence the expected exit time. The error analysis is shown for the one dimensional case and can be modified for higher dimensions. Three numerical examples are given. Two of the examples are two dimensional problems which have been given special attention by other investigators [2, 10] because of their computational difficulty.