Expansion of a filtration with a stochastic process: the information drift

TL;DR

This paper develops a method to compute the semimartingale decomposition when expanding a filtration with a stochastic process, ensuring Itô processes remain Itô processes, which is important for mathematical finance applications.

Contribution

It introduces a general approach to compute semimartingale decompositions in enlarged filtrations and provides conditions for Itô processes to remain Itô processes.

Findings

A new method to compute semimartingale decompositions in enlarged filtrations.

A sufficient condition for Itô processes to stay Itô processes after filtration expansion.

Implications for mathematical finance applications.

Abstract

When expanding a filtration with a stochastic process it is easily possible for semimartingale no longer to remain semimartingales in the enlarged filtration. Y. Kchia and P. Protter indicated a way to avoid this pitfall in 2015, but they were unable to give the semimartingale decomposition in the enlarged filtration except for special cases. We provide a way to compute such a decomposition, and moreover we provide a sufficient condition for It\^o processes to remain It\^o processes in the enlarged filtration. This has significance in applications to Mathematical Finance.