I am facing the following unusual problem: $Z_t$ is a pure drift process of the form $$ dZ_t = \kappa(X_t - Z_t) dt $$ where $X_t$ is another bounded process. I am interested in computing / upper-bounding the probability that it crosses a threshold $\alpha$ before a given horizon $T$, i.e. $$\mathbb{P}[\exists t \in (0,T) \text{ s.t. }Z_t \leq \alpha] << \epsilon$$ which in my case is a rather unlikely event. The fact that $X_t$ is bounded allows me to compute a minimal time $t_0$ such that the event is possible. Furthermore I can upper-bound the following probability for all times $t$ $$ \mathbb{P}[Z_t \leq \alpha] $$ relatively well using moments inequalities. However I do not see how to bound the main probability of interest. Maybe this is not the right way to approach the problem.

By advance, thanks for your suggestions.