First passage time density of an Ornstein–Uhlenbeck process with broken drift

We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.