Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes continuous almost everywhere with respect to the measure induced by $X_t$ (Wiener, correct?). I would be very grateful for any references!
2026-03-30 14:02:34.1774879354
Moving boundaries for Ornstein-Uhlenbeck processes
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Somewhat embarrassingly, it seems I found the answer (please correct me if I'm wrong).
An excellent book on the subject is Whitt: Stochastic Process Limits (Springer 2002). It provides nice intuition together with a useful toolbox for attacking these problems.
The keyword appears to be Skorokhod topologies. Elements in this space are continuous with jumps, or more precisely speaking, cadlag functions. A nice property of these functions is that they can be described as limits of sequences of piecewise constant functions with jumps. Whitt explains very nicely how properties such as first-passing time are analyzed by considering convergent sequences. It seems clear from this that cadlag boundaries will be fine, since OU processes are continuous (and thereby cadlag).