Résumé en anglais

Given a markovian Brownian martingale Z, we construct a Markov process X which is a martingale in its own filtration and satisfies 1 1 X =Z . We compute explicitly its semimartingale decomposition under both its own filtration and the filtration generated jointly by X and Z, so making a connection with (dynamic) enlargement of filtrations theory. As an application, we explicitely solve an equilibrium model with insider trading, that can be viewed as a generalization of Back and Pedersen's (Journal of Financial Markets 1, 1998) where stock price evolution exhibits a local volatility dynamics.