# First passage time of a pure drift process

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.

• Your process $Z_t$ admits the explicit representation $e^{-\kappa t}\Bigl(Z_0 + \kappa \int_0^t e^{\kappa s} X_s \, ds\Bigr)$. So without knowing the distribution of $X_t$ or some particular properties (as bounded away from zero), I do not expect a general result. – Stephan Sturm Nov 20 '14 at 5:07
• Thx. I have partial knowledge of the distribution of $X_t$ (the first $n$ moments), and $X_t \in (0,a)$. How would you proceed ? – user149575 Nov 20 '14 at 9:20
• I assume by moments you mean the moments of the marginals for some fixed t? I doubt that this will help much without knowing the transition probabilities of $X$. Could you be more precise how your process $X$ is given? – Stephan Sturm Nov 20 '14 at 17:16