Brownian motion and random walk perturbed at extrema

Let b(t) be Brownian motion. We shaw there is a unique adapted process x(t) which satisfies dx(t) = db(t) except when x(t) is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of the process. For some ranges of the parameters this is already known. We show that if a random walk close to b(t) is perturbed properly, its paths are close to those of x(t).