Functional Equations for the Stochastic Exponential

TL;DR

This paper explores stochastic analogs of the exponential functional equation, providing a martingale-based characterization of all solutions, thus advancing understanding of stochastic functional equations.

Contribution

It introduces a martingale framework to characterize solutions of stochastic exponential functional equations, extending classical deterministic results.

Findings

Martingale characterization of solutions

General solutions to stochastic exponential equations

Extension of classical functional equations to stochastic setting

Abstract

We consider stochastic versions of the Cauchy exponential functional equation and give a martingale characterization of the general solution.