Martingales with Independent Increments

TL;DR

This paper demonstrates that discrete-time martingales with atomless innovations can be decomposed into sums of martingales with independent increments, and continuous-time martingales from Brownian motion can be expressed as sums of Gaussian martingales.

Contribution

It introduces novel decompositions of martingales into sums of simpler martingales with independent or Gaussian increments.

Findings

Discrete-time martingales decompose into sums of independent increment martingales.

Continuous-time $L^2$ martingales from Brownian motion are sums of Gaussian martingales.

Provides a new perspective on the structure of martingales in stochastic processes.

Abstract

We show that a discrete time martingale with respect to a filtration with atomless innovations is the (infinite) sum of martingales with independent increments. For the continuous time filtration coming from Brownian Motion filtration, we show that every $L^2$ martingale is the sum of a series of Gaussian martingales.