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. For the Hurst parameter $H>1/2$ we establish equivalence of the measures, induced by mixed fBm process with stochastic drifts and derive an expression for the corresponding Radon-Nikodym derivative. For $H>3/4$ 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. This suggests a new insight on the regularization theorem due to P. Cheridito.