Conditioning of Gaussian processes and a zero area Brownian bridge

Maik Gorgens

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arXiv · 1302.4186

Conditioning of Gaussian processes and a zero area Brownian bridge

Maik Gorgens

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Abstract

We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on $[0,1]$ pinned down in 0 at time 1 and conditioned to have vanishing area spanned by the sample paths.

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Maik Gorgens. 2013-02-18. Conditioning of Gaussian processes and a zero area Brownian bridge. https://arxiv.org/abs/1302.4186

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