Mixed Fractional Brownian Motion

Abstract

We show that the sum of a Brownian motion and a non-trivial multiple of an independent fractional Brownian motion with Hurst parameter H P (0, 1] is not a semimartingale if H P (0, 1 2 ) ( 1 2 , 3 4 ], that it is equivalent to a multiple of Brownian motion if H 1 2 and equivalent to Brownian motion ifAs an application we discuss the price of a European call option on an asset driven by a linear combination of a Brownian motion and an independent fractional Brownian motion.