Projections in enlargements of filtrations under Jacod's hypothesis and examples.

Summary In this paper, we consider two kinds of enlargements of a Brownian filtration F: the initial enlargement with a random time τ , denoted by F (τ) , and the progressive enlargement with τ , denoted by G. We assume Jacod's equivalence hypothesis, that is, the existence of a positive F-conditional density for τ. Then, starting with the predictable representation of an F (τ)-martingale Y (τ) in terms of a standard F (τ)-Brownian motion, we consider its projection on G, denoted by Y G , and on F, denoted by y. We show how to obtain the coefficients which appear in the predictable representation property for Y G (and y) in terms of Y (τ) and its predictable representation. In the last part, we give examples of conditional densities.