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, except 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 minimum: we call it an alpha,beta-doubly perturbed Brownian motion. We complete some recent investigations by showing that for all permissible values of the parameters alpha, alpha and beta respectively, these equations have pathwise unique solutions, and these are adapted to the filtration of B.