First-Passage-Time in Discrete Time

We present a semi-closed form method of computing a first-passage-time (FPT) density for discrete time Markov stochastic processes. Our method provides exact solution for one-dimensional processes and approximations for higher dimensions. In particular, we show how to find an exact form of FPT for AR(1), and an approximate FPT for VAR(1). The method is valid for any type of innovation process if multi-period transition probabilities can be computed. It is intuitively straightforward, avoids the use of complex mathematical tools and therefore it is suitable for econometric applications. For instance, our method can be applied to form structural models of duration without the need to invoke simplistic continuous-time Brownian motion models. Finally, the method that we propose can be efficiently implemented by parallelization of computing tasks.

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