Adaptive prediction and reverse martingales

We study prediction problems for models where the underlying probability measure is not known. These problems are intimately connected with time reversal of Markov processes, and optimal predictors are shown to be characterized by being reverse martingales. For a class of diffusions we give a Feynman-Kac representation of the optimal predictor in terms of an associated complex valued diffusion and a concrete Wiener model is studied in detail. We also derive Cramér-Rao inequalities for the prediction error.prediction time reversal martingales diffusions point processes information inequalities