Martingale Transformations of Brownian Motion with Application to Functional Equations

Abstract

We describe the classes of functions f = (f (x), x ∈ R), for which processes f (W t ) − Ef (W t ) and f (W t )/Ef (W t ) are martingales. We apply these results to give a martingale characterization of general solutions of the quadratic and the D'Alembert functional equations. We study also the time-dependent martingale transformations of a Brownian Motion.