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A first passage time distribution for a discrete version of the Ornstein-Uhlenbeck process. (English) Zbl 1116.82305

Summary: 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.

MSC:

82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
60G50 Sums of independent random variables; random walks
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