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

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 γ-Hölder continuous for any γ < 2H − 3/2.