The fundamental martingale with applications to Markov Random Fields

Abstract: We consider collections of SDEs indexed by a graph. Each SDE is driven by an additive Gaussian noise and each drift term interacts with all other SDEs within the graph neighbourhood. We derive the fundamental martingale for a class of Gaussian processes and use this to prove a Girsanov type theorem. Further, we use this to construct a clique factorisation to prove that the law of the interacting SDEs forms a 2-Markov Random Field.