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First-Passage-Time Density and Moments of the Ornstein-Uhlenbeck Process

Luigi M. Ricciardi and Shunsuke Sato
Journal of Applied Probability
Vol. 25, No. 1 (Mar., 1988), pp. 43-57
DOI: 10.2307/3214232
Stable URL: http://www.jstor.org/stable/3214232
Page Count: 15
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First-Passage-Time Density and Moments of the Ornstein-Uhlenbeck Process
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Abstract

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.

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