In this work we combine 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 part of its sample path. This is done by considering special bases for the Gaussian space G generated by the process, which allows us to obtain an orthogonal basis for the Fock space of G such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily 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 bases elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process.
D. Alpay and A. Kipnis. Wiener-Chaos Approach to Optimal Prediction. Numerical Functional Analysis and Optimization, vol. 36 (2015), no. 10, 1286-1306.
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