  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.
