Transfer principle for 𝑛th order fractional Brownian motion with applications to prediction and equivalence in law

Sottinen, T.; Viitasaari, L.

doi:10.1090/tpms/1071

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

Authors: T. Sottinen and L. Viitasaari
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
MathSciNet review: 3824687
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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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