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

International audienceWe 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