0
$\begingroup$

If we consider a nice Ornstein Uhlenbeck process $d x (t) = - \gamma x(t) dt + \sigma d w (t)$ with $x(0) = x_0 \in (-L,L)$. Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener process.

Is the law of $\tau = \inf \{ t>0, |x(t)| = L \}$ the first hitting time of $\pm L$ by $x(t)$ known explicitly when $x_0 \neq 0$? When $x_0 = 0$, it is not a big issue.

Sorry if the solution is straightforward but it isn't clear to me.

Thanks for help. m.

$\endgroup$
  • $\begingroup$ I don't know the first reference, but this is known. Look up the first passage time of an Ornstein-Uhlenbeck process. $\endgroup$ – Douglas Zare Dec 10 '15 at 12:36
  • $\begingroup$ For example, see these notes: people.fas.harvard.edu/~sfinch/csolve/ou.pdf $\endgroup$ – Douglas Zare Dec 11 '15 at 11:54
  • $\begingroup$ Thanks a lot for the two comments. I will go through these notes and be back soon. m. $\endgroup$ – megaproba Dec 14 '15 at 10:09

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.