On martingale approximation of adapted processes

TL;DR

This paper investigates how the existence of martingale approximations for stationary processes depends on the choice of filtration, revealing that such approximations are not always consistent across different filtrations.

Contribution

It demonstrates that the existence of martingale approximations varies with the filtration, providing examples where approximations exist or fail depending on the filtration used.

Findings

Martingale approximation depends on the filtration choice.

Existence of approximation can differ between natural and larger filtrations.

Examples show processes with and without martingale approximations under various filtrations.

Abstract

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to wich it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.