Graphical modeling of stochastic processes driven by correlated noise

We study a class of graphs that represent local independence structures in stochastic processes allowing for corre-lated noise processes. Several graphs may encode the same local independencies and we characterize such equiv-alence classes of graphs. In the worst case, the number of conditions in our characterizations grows superpolyno-mially as a function of the size of the node set in the graph. We show that deciding Markov equivalence of graphs from this class is coNP-complete which suggests that our characterizations cannot be improved upon substantially. We prove a global Markov property in the case of a multivariate Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.