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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 constants 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.

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  • $\begingroup$ 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
    Commented Dec 10, 2015 at 12:36
  • $\begingroup$ For example, see these notes: people.fas.harvard.edu/~sfinch/csolve/ou.pdf $\endgroup$
    – Douglas Zare
    Commented Dec 11, 2015 at 11:54
  • $\begingroup$ Thanks a lot for the two comments. I will go through these notes and be back soon. m. $\endgroup$
    – megaproba
    Commented Dec 14, 2015 at 10:09

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It turns out that finding its density is still open as mentioned here: On the First Hitting Time Density of an Ornstein-Uhlenbeck Process.

The Laplace transform has been computed though as mentioned there too.

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