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 explicitly its semimartingale decomposition under both its own
filtration $\cF^X$ and the filtration $\cF^{X,Z}$ jointly generated by $X$ and
$Z$. Our construction is heavily based on parabolic PDE's and filtering
techniques.