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

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Abstract

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 Hölder 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.

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Authors and Affiliations

  1. Laboratory of Mathematical Physics, The Rockefeller University, 10021, New York, NY, USA

    Thibaud Taillefumier & Marcelo O. Magnasco

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  1. Thibaud Taillefumier
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  2. Marcelo O. Magnasco
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Correspondence to Thibaud Taillefumier.

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Taillefumier, T., Magnasco, M.O. A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Hölder Continuous Boundaries. J Stat Phys 140, 1130–1156 (2010). https://doi.org/10.1007/s10955-010-0033-6

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