Filtration shrinkage, the structure of deflators, and failure of market completeness

TL;DR

This paper investigates how market deflators behave under reduced information, revealing that projections can lose completeness and fail to span all deflators, especially in continuous-path models.

Contribution

It provides a detailed analysis of the structure of projected local martingale deflators and explains the failure of market completeness under filtration shrinkage using a Bayesian filtering approach.

Findings

Projected deflators are local martingale deflators in smaller markets.

Filtration shrinkage can cause deflators to become strict supermartingales.

Market completeness may fail under reduced information.

Abstract

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.