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Theory of Probability and Mathematical Statistics

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Transfer principle for $ n$th order fractional Brownian motion with applications to prediction and equivalence in law


Authors: T. Sottinen and L. Viitasaari
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 199-216
MSC (2010): Primary 60G22; Secondary 60G15, 60G25, 60G35, 60H99
DOI: https://doi.org/10.1090/tpms/1071
Published electronically: August 19, 2019
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Abstract: The $ n$th order fractional Brownian motion was introduced by Perrin et al.in IEEE Transactions on Signal Processing 49 (2001), no. 5, 1049-1059. It is the (up to a multiplicative constant) unique self-similar Gaussian process with the Hurst index $ H \in (n-1,n)$, having $ n$th order stationary increments. We provide a transfer principle for the $ n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $ n$th order fractional Brownian motion and then represent the $ n$th order fractional Brownian motion by using the Brownian motion in a nonanticipative way so that the filtrations of the $ n$th order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the $ n$th order fractional Brownian motion and also a representation formula for all Gaussian processes that are equivalent in law to the $ n$th order fractional Brownian motion.


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Additional Information

T. Sottinen
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland
Email: tommi.sottinen@iki.fi

L. Viitasaari
Affiliation: Department of Mathematics and Statistics, University of Helsinki, Helsinki, P.O. Box 68, FIN-00014 University of Helsinki, Finland
Email: lauri.viitasaari@iki.fi

DOI: https://doi.org/10.1090/tpms/1071
Keywords: Fractional Brownian motion, stochastic analysis, transfer principle, prediction, equivalence in law
Received by editor(s): January 23, 2018
Published electronically: August 19, 2019
Additional Notes: We thank the anonymous referee for comments that greatly improved the paper
Article copyright: © Copyright 2019 American Mathematical Society