On the first passage time for autoregressive processes

The first passage time problem for an autoregressive process AR(p) is examined. When the innovations are gaussian, the determination of the first pas- sage time probability distribution is closely related to computing a multidimensional integral of a suitable gaussian random vector, known in the literature as orthant prob- ability. Recursive equations involving the first passage time probability distribution are given and a numerical scheme is proposed which takes advantage of the recursion. Compared with the existing procedures in the literature, the algorithm we propose is computationally less expensive and reaches a very good accuracy. The accuracy is tested on some closed form expressions we achieve for special choices of the AR(p) parameters.