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

摘要

In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the "future" of a stochastic process, and seeks its best predictor ( in the L-2-sense). In this paper we treat a variant of the prediction problem, whereby we seek the "most predictable" non-trivial functional of the future and its best predictor; we refer to such a pair ( if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite "past" and "future" intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous- time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.