Wiener-Chaos Approach to Optimal Prediction

TL;DR

This paper develops a Wiener chaos expansion method for optimal prediction of Gaussian processes with stationary increments, including fractional Brownian motion, by constructing a special basis for the Gaussian space.

Contribution

It introduces a novel basis construction for the Fock space that enables chaos expansion of conditioned variables in Gaussian processes.

Findings

Derived chaos expansion for conditioned Gaussian variables.

Constructed a basis for Gaussian processes with stationary increments.

Provided a prediction formula for fractional Brownian motion.

Abstract

The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by constructing a special basis for the Fock space of the Gaussian space generated by the process, such that each basis element is either measurable or independent with respect to the given samples. This property of the basis allows us to derive the chaos expansion of a random variable conditioned on part of the sample path. We provide a general method for the construction of such basis when the underlying process is Gaussian with stationary increment. We evaluate the basis elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process.