Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z , we build a process X which is a martingale in its own filtration and satisfies X 1 = Z 1 . We call X a dynamic bridge, because its terminal value Z 1 is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F X and the filtration F X , Z jointly generated by X and Z . Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.