Some calculations for doubly perturbed Brownian motion

In the present paper we compute the laws of some functionals of doubly perturbed Brownian motion, which is the solution of the equation Xt=Bt+α sups⩽t Xs+β infs⩽t Xs, where α,β<1, and B is a real Brownian motion. We first show that the process obtained by juxtaposing the positive (resp. negative) excursions of this solution depends only on α (resp. β). Moreover, these two processes are independent. As a consequence of this splitting we compute, by direct calculations, the law of the occupation time in [0,∞) and we specify the joint distribution of the time and position at which doubly perturbed Brownian motion exits an interval.