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 X1 = Z1. We call X a dynamic bridge, because its terminal value Z1 is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration FX and the filtration FX,Z jointly generated by X and Z. Our construction is heavily based on parabolic PDE’s 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 Back and Pedersen’s [3], where insider’s additional information evolves over time. Key-words: Markovian bridges, martingale problem, nonlinear filtering, parabolic PDE’s, equilibrium, insider trading. AMS classification (2000): 60G44, 60H05, 60H10, 93E11 JEL classification: D82, G14