×

First-passage-time density and moments of the Ornstein-Uhlenbeck process. (English) Zbl 0651.60080

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

MSC:

60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI