Martingale spaces and representations under absolutely continuous changes of probability

TL;DR

This paper investigates the relationship between martingale spaces under different probability measures and demonstrates that the martingale representation property remains stable under locally absolutely continuous changes of probability, using a minimal and constructive approach.

Contribution

It introduces a general, minimal, and constructive framework for analyzing martingale spaces under locally absolutely continuous probability changes, extending existing theories.

Findings

MRP is stable under locally absolutely continuous probability changes

Provides a simple example illustrating the theory's applicability

Extends the understanding of martingale representations in general settings

Abstract

In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.