The fundamental martingale with applications to Markov Random Fields

TL;DR

This paper develops a fundamental martingale for Gaussian processes in graph-indexed SDEs, enabling a Girsanov theorem and showing the law of these SDEs forms a 2-Markov Random Field, with applications to Markov Random Fields.

Contribution

It introduces a fundamental martingale for Gaussian processes in graph-structured SDEs and proves the law forms a 2-Markov Random Field, extending stochastic analysis tools.

Findings

Derived the fundamental martingale for Gaussian processes in graph SDEs.

Proved a Girsanov type theorem for these processes.

Showed the law of the SDEs forms a 2-Markov Random Field.

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.