Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation [GRAPHICS] where B is a Brownian motion, behaves like a reflected Brownian motion, exc ept when it attains a new maximum: we call it an alpha-perturbed reflected Brownian motion. Similarly any solution of [GRAPHICS] behaves like a Brownian motion except when it attains a new maximum or mini mum: we call it an alpha,beta-doubly perturbed Brownian motion. We complete some recent investigations by showing that for all permissible values of t he parameters alpha, alpha and beta respectively, these equations have path wise unique solutions, and these are adapted to the filtration of B.