  In this paper, we introduce different concepts of Granger non-causality and
contemporaneous uncorrelation for stationary continuous-time processes to model
the different dependencies between the component series of multivariate time
series models. Several equivalent characterisations for the different
definitions are given, in particular by linear projections. We then define two
mixed graphs based on different definitions of causality and contemporaneous
uncorrelation, the (mixed) causality graph and the local (mixed) causality
graph, to visualise and to analyse the different dependencies in stationary
continuous-time processes. In these graphs, the components of the process are
represented by vertices, directed edges between the vertices indicate causal
influences and undirected edges indicate contemporaneous uncorrelation between
the component processes. Further, we introduce various notions of causal Markov
properties in analogy to Eichler (2012), which relate the different dependence
structures of subprocesses, and we derive sufficient criteria for the (local)
causality graph to satisfy them. Finally, as an example, the popular
multivariate continuous-time AR (MCAR) processes satisfy our assumptions. For
MCAR processes we show that the (local) causality graphs can be characterised
explicitly by the model parameters.
