A simple method to calculate first-passage time densities of non-smooth processes

Numerous applications all the way from biology and physics to economics depend on the density of first crossings over a boundary. Motivated by the lack of analytical tools for computing first-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the Independent Interval Approximation (IIA). We generalise previous formulations of the IIA to handle non-smooth processes, and derive a closed form expression for the FPTD in Laplace and $z$-transform space for arbitrary boundary and starting points in one dimension. We focus on Markov processes for which the IIA is exact. To apply our equations, we calculate the FPTD in two cases: the Ornstein-Uhlenbeck process and the discrete time Brownian walk. Our results are in good agreement with Langevin dynamics simulations.

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