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

TL;DR

The paper analyzes the regularity of the drift in the Doob–Meyer decomposition of the mixed fractional Brownian motion $X_t=B^H_t+B_t$ in the semimartingale regime $H>3/4$. It provides an innovation-based representation of the drift via a kernel $L(r,s)$ that solves a weakly singular integral equation and develops auxiliary estimates for such equations. The main result shows that the drift’s derivative is differentiable with a continuous modification locally $\gamma$-Hölder for every $\gamma<2H-3/2$, with $H=1$ recovering the known linear case. These findings have implications for modeling volatility and market impact in finance using mixed fBm, clarifying the smoothness of the price-trajectory drift in this framework.

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 is greater than $3/4$. The question in the title is motivated by recent findings in quantitative finance. In this note, we find that the drift in its Doob-Meyer decomposition has derivative which is $γ$ Hölder for any $γ< 2H-3/2$.

Theorems & Definitions (4)

• Theorem 1.1
• Remark 2.1
• Lemma 2.2
• proof