June 18, 2002 16:30-18:00, TU FH, Turm A, 6. Stock, Seminarraum 107 Abstract: (1) We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, and the conditioning of a standard Brownian motion by its first hitting time of a given level. (2) As an application, we introduce the notion of weak information on a complete or incomplete market, and we give a "quantitative" value to this weak information. (3) As the end of the talk, we will go one step further by considering a flow of information: as the price process is revealed, the weak anticipation is updated.