Abstract
We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.
Highlights
The stochastic analysis of Gaussian processes that are not semimartingales is challenging
We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations
One way to overcome the challenge is to represent the Gaussian process under consideration, X, say, in terms of a Brownian motion, and develop a transfer principle so that the stochastic analysis can be done in the “Brownian level” and transferred back into the level of X
Summary
The stochastic analysis of Gaussian processes that are not semimartingales is challenging. More general Gaussian processes have been studied in the already mentioned work by Alos et al [1] They considered Gaussian Volterra processes where the kernel satisfies certain technical conditions. Their results cover fractional Brownian motion with Hurst parameter H > 1/4.
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