Functional equations and martingales

TL;DR

This paper explores the connection between functional equations and martingales, demonstrating that solving certain functional equations is equivalent to transforming Brownian motion into a martingale through specific functions.

Contribution

It establishes a novel equivalence between solving classical functional equations and constructing martingales from Brownian motion via space transformations.

Findings

Functional equations relate to martingale transformations.

Solutions involve space-transformations of Brownian motion.

Provides a unified approach to classical functional equations.

Abstract

We consider functional equations (Cauchy's, Abel's and some other functional equations) and show that to find general solution of these equations is equivalent to establish that a space-transformation of a Brownian Motion by suitable function (or functions) is a martingale.