Optimal Transformations for Prediction in Continuous-Time Stochastic Processes

In the classical Wiener-Kolmogorov prediction problem, one fixes a functional of the “future” and seeks its best predictor (in the L2-sense). In this paper we treat a variant of this problem, whereby we seek the “most predictable” non-trivial functional of the future and its best predictor. In contrast to the Wiener-Kolmogorov problem, our problem may not have solutions, and if solutions exist, they might not be unique. We prove the existence of solutions for linear functionals under appropriate conditions on the spectral function of weakly stationary, continuous-time processes.