On the first positive and negative excursion exceeding a given length

For a one-dimensional diffusion process $X$, we derive the Laplace transform and the moments of the first time at which the age of an excursion above (or below) the level $x$ is longer than $u$. The result is then illustrated for diffusion processes that are found relevant in applications. In the context of pricing Parisian options, the Brownian motion and the Geometric Brownian motion are considered and the Laplace transform can be made explicit and explicit expression for the moments can be derived. In the context of neuronal modeling, the Ornstein--Uhlenbeck process and the Cox--Ingersoll--Ross process are considered and the Laplace transform and the moments must be approximated by numerical inversion.