Optimal transformations for prediction in continuous-time stochastic processes: finite past and future (Q1773989)

The authors treat a variant of the prediction problem, slightly different from the classical Wiener-Kolmogorov one. Namely, the best predictor and the most predictable functional are considered. This pair will be the so called optimal transformation for prediction. It is shown that the problem may not have a solution, or it may have multiple solutions. The existence of the optimal transformation under some appropriate conditions on the spectral density is proved. Some interesting open problems here arise. The paper has three parts. In the introduction the problem is presented. Section 2 contains the proof of the existence of the solution under some conditions. In Section 3 the case when the spectral density is rational is analysed. An explicit construction of the optimal transformation in terms of the coefficients of the polynomials which compose the rational spectral density is obtained.