On the sub-mixed fractional Brownian motion

TL;DR

This paper investigates a new stochastic process formed by combining Brownian motion with sub-fractional Brownian motion, analyzing its properties, and identifying conditions under which it is not a semi-martingale.

Contribution

It introduces and studies the sub-mixed fractional Brownian motion, revealing its characteristics and its position between known processes, and determines when it lacks semi-martingale properties.

Findings

S^H can be viewed as an intermediate process between sub-fractional and mixed fractional Brownian motions.

The process's main properties are characterized, including its non semi-martingale conditions.

The process bridges the gap between different types of fractional Brownian motions.

Abstract

Let ${S_t^H, t \geq 0} $ be a linear combination of a Brownian motion and of an independent sub-fractional Brownian motion with Hurst index $0 < H < 1$. Its main properties are studied and it is shown that $S^H $ can be considered as an intermediate process between a sub-fractional Brownian motion and a mixed fractional Brownian motion. Finally, we determine the values of $H$ for which $S^H$ is not a semi-martingale.