Ricciardi, Luigi M.; Sato, Shunsuke First-passage-time density and moments of the Ornstein-Uhlenbeck process. (English) Zbl 0651.60080 J. Appl. Probab. 25, No. 1, 43-57 (1988). Formulas are derived expressing the first-passage-time probability density function \(g(t,S| x_ 0)\) through the boundary S and the n th moments \[ t_ n(S| x_ 0)=\int^{\infty}_{0}t^ ng(t,S| x_ 0)dt,\quad n=1,2,..., \] for the Ornstein-Uhlenbeck process with drift -x/\(\theta\) \((\theta >0)\) and infinitesimal variance \(\mu\). It is proved that for \(\theta =1\), \(\mu =2\) and large S \[ g(t,S| x_ 0)\sim g(S)\exp (-g(S)t),\quad and\quad t_ n(S| x_ 0)\sim n![g(S)]^ n,\quad n=1,2,..., \] where \(g(z)=2(2\pi)^{-1/2}\exp (-z^ 2/2)\). Reviewer: B.Grigelionis Cited in 1 ReviewCited in 58 Documents MSC: 60J60 Diffusion processes Keywords:first-passage-time; Ornstein-Uhlenbeck process PDFBibTeX XMLCite \textit{L. M. Ricciardi} and \textit{S. Sato}, J. Appl. Probab. 25, No. 1, 43--57 (1988; Zbl 0651.60080) Full Text: DOI