How smooth is the drift of the mixed fractional Brownian motion?

Abstract

The mixed fractional Brownian motion - the sum of independent fractional and standard Brownian motions - is known to be a semimartingale if the Hurst exponent $H$ of its fractional component satisfies $H>3/4$. The question posed in the title is motivated by recent findings in quantitative finance. In this note, we show that the drift in its Doob-Meyer decomposition has a derivative that is $\gamma$-Hölder continuous for any $\gamma <2H-3/2$.