Some calculations for doubly perturbed Brownian motion

In the present paper we compute the laws of some functionals of doubly pert urbed Brownian motion, which is the solution of the equation X-t = B-t + al pha sup(s less than or equal to t)X(s) + beta inf(s less than or equal to t )X(s), where alpha, beta < 1, and B is a real Brownian motion. We first sho w that the process obtained by juxtaposing the positive (resp. negative) ex cursions of this solution depends only on alpha (resp. beta). Moreover, the se two processes are independent. As a consequence of this splitting we com pute, by direct calculations, the law of the occupation time in [0, infinit y) and we specify the joint distribution of the time and position at which doubly perturbed Brownian motion exits an interval. 0 2000 Published by Els evier Science B.V. All rights reserved.