Abstract
We consider the initial and progressive enlargements of a Brownian filtration with a random time, that is, a strictly positive random variable. We assume Jacod’s equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the Brownian filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged Brownian filtration, we derive explicit expressions for the components which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the Brownian filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the Brownian filtration.
Highlights
In this paper, we consider the initial enlargement of a Brownian filtration F with a strictly positive absolutely continuous random variable τ, denoted by F(τ)
Projections of martingales in enlargements of Brownian filtrations below for details), which implies that there exists an F(τ)-martingale enjoying the predictable representation property with respect to F(τ) and a pair of G-martingales enjoying the predictable representation property with respect to G
We study the relationship between the representation of martingales in the initially enlarged filtration and the various optional projections
Summary
We consider the initial (resp. progressive) enlargement of a Brownian filtration F (called hereafter the reference filtration) with a strictly positive absolutely continuous random variable τ (called hereafter a random time), denoted by F(τ) Progressively) enlarged filtration and the various optional projections. We show that the set of equivalent martingale measures in the model with the progressively enlarged filtration G is essentially larger than the one obtained by means of the optional projections on G of the Radon-Nikodym densities in the model with the initially enlarged filtration F(τ)
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