Foundations of the Prediction Process

This book aims to present a unified treatment of the prediction process approach to continuous time stochastic processes. The underlying idea is that there are two kinds of time: stationary physical time and the moving observer's time. By developing this theme, Professor Knight develops a theory of stochastic processes whereby two processes are considered rather than one, which coexist on the same probability space. In this way, the observer's process is strongly Markovian. Consequently, any measurable stochastic process of real parameter may be regarded as a homogeneous strong Markov process in an appropriate setting. This leads to a unifying principle for the representation of general processes in terms of martingales, which facilitates the prediction of their properties. Whilst the ideas are advanced, the methods are reasonably elementary and should be accessible to a reader having a basic knowledge of measure theory, functional analysis, stochastic integration, and probability, on the level of the convergence theorem for positive super-martingales.Table of ContentIntroduction; The measure - theoretic prediction process; Topological considerations and prediction space: application to Markov processes; Gaussian processes and the prediction process in the wide sense; A classification of measurable processes; Prediction of measurable processes; Application to concrete examples, and ramifications of the theory; Appendix; References; Index.