A Characterization of Adapted Distribution

The notion of adapted distribution of a stochastic process was introduced in a recent paper of Hoover and Keisler. Here we give a simple characterization of this notion in terms of filtration embeddability. This characterization allows us to show that for a local martingale M for which some ordinary stochastic differential equation Xt= ∫t 0f(s, Xs) dMsadmits sufficient nonuniqueness in law of the solutions X, the class of possible joint laws of (M, X) determines the adapted law of M.