A martingale decomposition of discrete Markov chains

- We consider a multivariate time series given from a discrete Markov chain.
- Its martingale decomposition is derived, with all terms given in closed form.
- The decomposition is analogous to the Beveridge-Nelson decomposition.
- Decomposition has three terms: a persistent, a transitory, and a deterministic trend.
- The autocovariance structure across all terms is fully characterized.

We consider a multivariate time series whose increments are given from a homogeneous Markov chain. We show that the martingale component of this process can be extracted by a filtering method and establish the corresponding martingale decomposition in closed-form. This representation is useful for the analysis of time series that are confined to a grid, such as financial high frequency data.