Causality and stochastic realization

The study of Granger causality has been mainly preoccupied with time series. We will instead concentrate on continuous time processes. Many systems to which it is natural to apply tests of causality take place in continuous time. For example, this is generally the case within economy. In the first part of this paper, we give a generalization of a causality relationship “G is a cause of E within H” which (in terms of σ-algebras) was first given in [4] and which is based on Granger’s definition of causality [2]. In the second part, we relate concepts of causality to the stochastic realization problem. The approach adopted in this paper is that of [3]. However, since our results do not depend on probability distribution, we deal with arbitrary Hilbert spaces instead of those generated by Gaussian processes. We suppose that it is known that a stochastic dynamic system (s.d.s.) S1 with known outputs H causes, in a certain sense, behaviour of some other s.d.s. S2, whose states (or some information about them) E are given. The main problem, to be formulated more precisely below, is to determine Markovian representations G (as a state space of an s.d.s. S1) for a family H such that family G provides exactly the same amount of information about the family H as family E (see Definition 2.10). It is clear that all the results of this paper can be extended on the σ-algebras generated by finite-dimensional Gaussian random variables. But, in the case that σ-algebras are arbitrary, the extensions of the proofs of this paper are nontrivial because one cannot take an orthogonal complement with respect to a σ-algebra as one can with respect to subspaces in a Hilbert space.