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# Theory of Probability and Mathematical Statistics

## Transfer Principle for MakeLowercase n-th order Fractional Brownian Motion with Applications to Prediction and Equivalence in Law

### T. Sottinen, L. Viitasaari

Abstract: The n-th order fractional Brownian motion was introduced by Perrin et al. [13]. It is the (up to a multiplicative constant) unique self-similar Gaussian process with the Hurst index Hâˆˆ(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 non-anticipative 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.

Keywords: Fractional Brownian motion, stochastic analysis, transfer principle, prediction, equivalence in law.

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