Conditional hitting time distribution for OU process between two fixed barriers

Question

For an the OU process $\mathrm dX_t=-\omega X_t\,\mathrm dt+\omega\mathrm dW_t$ with $X_0=x_0$ and $L<x_0<U$ let $$ \tau=\inf\{t>0:X_t\notin(L,U)\} $$ denote the hitting time of either barrier $L$ or barrier $U$. The survival probability $P(x_0,t)=\mathsf P(\tau>t)$ can be expressed as a boundary value problem in terms of the infinitesimal generator $L_t$ $$ \begin{cases} L_t P(x_0,t)=\partial_t P(x_0,t)\\ P(x_0,0) = 1_{(L,U)}\\ P(L,0)=0\\ P(U,0)=0. \end{cases} $$

Is anyone aware of a similar formulation for the survival probability conditioned on the process hitting one of the barriers? Are there any numerical algorithms for computing the hitting time distribution conditioned on the process hitting a specified barrier?

The author of this post does provide a solution for the probability of hitting either boundary.

Answer 1

Conditioning on hitting the left barrier before the right one amounts to weighting the measure by the martingale $P(X_{t\wedge\tau})/P(x_0),$ where $P(x)$ denotes the probability to exit on the left starting from $x$. By Girsanov theorem, we have $dW_t=(\log P)'(X_t)\,dt+d\tilde{W}_t,$ where $\tilde{W}_t$ is a Brownian motion under the weighted measure. Therefore, the survival probability satisfies the same boundary value problem as in the OP, but with the modified generator $$\tilde{L}_x=\left(-\omega x+(\log P)'(x)\right)\partial_x+\frac{\omega^2}{2}\partial^2_x.$$