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

TL;DR

This paper introduces a straightforward method using the Independent Interval Approximation to compute first-passage time densities for non-smooth processes, applicable to various complex systems.

Contribution

It generalizes the IIA for non-smooth processes and derives a closed-form FPTD expression in Laplace and z-transform spaces for Markov processes.

Findings

Accurately computes FPTDs for Ornstein-Uhlenbeck process

Validates results with Langevin dynamics simulations

Provides a versatile analytical tool for complex stochastic processes

Abstract

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.