A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

TL;DR

The paper introduces a novel filtration augmentation method tailored for constructing probability measures linked to strict local martingales, enabling classical stochastic analysis results on specific stochastic intervals.

Contribution

It proposes a new filtration augmentation technique compatible with strict local martingale measure changes, bridging measure construction and classical stochastic analysis.

Findings

Enables construction of measures associated with strict local martingales

Preserves classical stochastic analysis results on certain intervals

Provides a framework for measure changes in stochastic processes

Abstract

In this note we introduce a new kind of augmentation of filtrations along a sequence of stopping times. This augmentation is suitable for the construction of new probability measures associated to a positive strict local martingale as done in \cite{split1}, while it is on the other hand rich enough to make classical results from stochastic analysis hold true on some stochastic interval of interest.