A first passage problem for a bivariate diffusion process: numerical solution with an application to neuroscience

TL;DR

This paper develops a numerical method to analyze the first passage time of a component in a bivariate diffusion process, with applications to neuroscience, by solving a new integral equation.

Contribution

It introduces a novel integral equation for the first passage time density and proposes a convergent numerical algorithm for its solution.

Findings

Validated the method on integrated Brownian Motion

Applied the approach to integrated Ornstein-Uhlenbeck process

Discussed a neuroscience-related model

Abstract

We consider a bivariate diffusion process and we study the first passage time of one component through a boundary. We prove that its probability density is the unique solution of a new integral equation and we propose a numerical algorithm for its solution. Convergence properties of this algorithm are discussed and the method is applied to the study of the integrated Brownian Motion and to the integrated Ornstein Uhlenbeck process. Finally a model of neuroscience interest is also discussed.