We will present some new results on Kyle-Back equilibrium models. In particular, we will consider the case of an insider observing a dynamic signal, modelled as a Brownian Markovian martingale so that finding an equilibrium reduces to constructing a ‘dynamic’ Markov bridge in the sense that its terminal value is not known in advance, being the terminal value of insider’s signal. We think that such a construction is also interesting from a purely probabilistic point of view. The end of the talk will be devoted to discussing a Merton-type model for credit risk with insider trading. This talk is based on joint works with Umut Cetin, Albina Danilova and Alessandro Sbuelz.