Martingales Associated with Finite Markov Chains

Abstract

In a recent paper, [1], Phillipe Biane introduced martingales M ^k associated with the different jump ‘sizes’ of a time homogeneous, finite Markov chain and developed homogeneous chaos expansions. It has long been known that the Kolmogorov equation for the probability densities of a Markov chain gives rise to a canonical martingale M. The modest contributions of this note, are that working with a non-homogeneous chain, we relate Biane’s martingales M ^k to M, calculate the quadratic variation of M and thereby that of the M ^k. In addition, square field identities are obtained for each jump size.