Martingale Transformations of Brownian Motion with Application to Functional Equations

TL;DR

This paper characterizes functions that transform Brownian motion into martingales and applies these results to solve functional equations, providing new insights into stochastic processes and their applications.

Contribution

It introduces a classification of functions transforming Brownian motion into martingales and applies this to solve quadratic and D'Alembert functional equations.

Findings

Identifies classes of functions making transformed Brownian motions martingales

Provides a martingale-based characterization of solutions to key functional equations

Studies time-dependent martingale transformations of Brownian motion

Abstract

We describe the classes of functions $f=(f(x), x\in R)$, 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.