Level-crossing probabilities and first-passage times for linear processes

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1, 1) models and the mean first passage time for AR(P) processes. We also extend Novikov's martingale approach to ARMA(p, q) processes. Numerical schemes are used to solve the integral equations for specific examples.