A first passage problem for a bivariate diffusion process: Numerical solution with an application to neuroscience when the process is Gauss–Markov

We consider a bivariate Gauss–Markov process and we study the first passage time of one component through a constant boundary. We prove that its probability density function 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 discussed.