Mixed fractional Brownian motion: the filtering perspective

The paper presents an alternative approach to studying the properties of the mixed fractional Brownian motion (fBm) and related models, based on the filtering theory of Gaussian processes. The results shed light on its semimartingale structure and lead to a number of useful absolute continuity relations. We establish equivalence of the measures, induced by mixed fBm process with stochastic drifts, and derive the corresponding expression for Radon-Nikodym derivative. For the Hurst parameter H>3/4 we obtain a representation of the mixed fBm as a diffusion type process in its own filtration and derive a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H<1/4, we prove equivalence to the fractional part and find a formula for the corresponding derivative. These findings provide a new insight on the regularization theorem due to P. Cheridito and its generalization due to H. van Zanten.