# hitting time of an Ornstein-Ulhenbeck

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.

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