Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a$a$. As a consequence, we obtain the probabilities that a drawdown of size a$a$ precedes a drawup of size b$b$ and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross process.