A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Holder Continuous Boundaries

Even for simple diffusion processes, treating first-passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly. Here, we present a novel numerical method that is faster and has more tightly controlled accuracy. Our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non-negative homogeneously Holder continuous boundaries by Gauss-Markov processes. These include the Ornstein-Uhlenbeck process underlying the ubiquitous "leaky integrate-and-fire" model of neuronal excitation. Our method evaluates discrete points in a sample path exactly, and refines this representation recursively only in regions where a passage is rigorously estimated to be probable (e. g. when close to the boundary). As a result, for a given temporal accuracy in the location of the first passage time, our method is orders of magnitude faster than direct forward integration such as Euler or stochastic Runge-Kutta schemata. Moreover, our algorithm rigorously bounds the probability that such crossings are not true first-passage times.