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 its semimartingale decomposition explicitly under both its own filtration...

Every submartingale S of class D has a unique Doob–Meyer decomposition S=M+A, where M is a martingale and A is a predictable increasing process starting at 0.