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

TL;DR

This paper investigates the regularity of the drift in the mixed fractional Brownian motion's Doob-Meyer decomposition, revealing it is $ ext{γ}$-Hölder continuous for any $ ext{γ} < 2H - 3/2$, especially when $H > 3/4$.

Contribution

It provides a new regularity result for the drift's derivative in the mixed fractional Brownian motion, linking it to the Hurst exponent $H$.

Findings

Drift derivative is $ ext{γ}$-Hölder continuous for $ ext{γ} < 2H - 3/2$.

The result applies when the Hurst exponent $H > 3/4$.

Enhances understanding of the regularity properties of 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 $\gamma$-H\"older continuous for any $\gamma < 2H - 3/2$.