  We discuss the finite sample theoretical properties of online predictions in
non-stationary time series under model misspecification. To analyze the
theoretical predictive properties of statistical methods under this setting, we
first define the Kullback-Leibler risk, in order to place the problem within a
decision theoretic framework. Under this framework, we show that a specific
class of dynamic models -- random walk dynamic linear models -- produce exact
minimax predictive densities. We first show this result under Gaussian
assumptions, then relax this assumption using semi-martingale processes. This
result provides a theoretical baseline, under both non-stationary and
stationary time series data, for which other models can be compared against. We
extend the result to the synthesis of multiple predictive densities. Three
topical applications in epidemiology, climatology, and economics, confirm and
highlight our theoretical results.
