A first passage time distribution for a discrete version of the Ornstein-Uhlenbeck process

The probability of the first entrance to the negative semi-axis for a one-dimensional discrete Ornstein-Uhlenbeck (O-U) process is studied in this work. The discrete O-U process is a simple generalization of the random walk and many of its statistics may be calculated using essentially the same formalism. In particular, the case in which Sparre-Andersen's theorem applies for normal random walks is considered, and it is shown that the universal features of the first passage probability do not extend to the discrete O-U process. Finally, an explicit expression for the generating function of the probability of first entrance to the negative real axis at step n is calculated and analysed for a particular choice of the step distribution.