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.
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