2015/01/09 by Kerry Fendick, Fendick, Kerry
Computer Science · Environmental Science · Mathematics · #60G25 #FOS: Mathematics #G.3 #Gaussian Processes and Bayesian Inference #Hydrology and Drought Analysis #Probability (math.PR) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1501.02229
openalex publication_date 2015/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The solution to a multivariate linear Stochastic Differential Equation (SDE) with constant initial state is well known to be a Gaussian Markov process, but its covariance kernel involves the solution to an integral equation in the general case. We show that the covariance kernel has a simpler semi-parametric form for families of such solutions representing increments of a common process. We also show that a covariance kernel of a particular parametric form is necessary and sufficient for a solution to possess stationary increments and for a Gaussian process, in considerable generality, to have stationary increments and the Markov property. For a discretely sampled Gaussian process with such a parametric kernel, we derive closed-form expressions for unique maximum likelihood estimators of the parameter matrices that are unbiased, jointly sufficient, and easily computed regardless of the dimension. Using those estimators, we also derive closed-form expressions for posterior moments useful for forecasting.