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Zbl 0651.60080
Ricciardi, Luigi M.; Sato, Shunsuke
First-passage-time density and moments of the Ornstein-Uhlenbeck process. (English)
[J] J. Appl. Probab. 25, No.1, 43-57 (1988). ISSN 0021-9002

Formulas are derived expressing the first-passage-time probability density function $g(t,S\vert x\sb 0)$ through the boundary S and the n th moments $$ t\sb n(S\vert x\sb 0)=\int\sp{\infty}\sb{0}t\sp ng(t,S\vert x\sb 0)dt,\quad n=1,2,..., $$ 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 $$ g(t,S\vert x\sb 0)\sim g(S)\exp (-g(S)t),\quad and\quad t\sb n(S\vert x\sb 0)\sim n![g(S)]\sp n,\quad n=1,2,..., $$ where $g(z)=2(2\pi)\sp{-1/2}\exp (-z\sp 2/2)$.
[B.Grigelionis]

MSC 2000:: *60J60 Diffusion processes

Keywords: first-passage-time; Ornstein-Uhlenbeck process

Cited in: Zbl 0954.60067

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