We study the semimartingale properties of the generalised fractional Brownian motion (GFBM) introduced by Pang and Taqqu (High Freq. 2:95–112, 2019) and discuss applications of GFBM and its mixtures to financial asset pricing. The GFBM $X$ is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the regions of these two parameter values where GFBM is a semimartingale with respect to its natural filtration $\mathbb{F}^{X}$ . We next study the mixed process $Y$ made up of an independent BM and a GFBM and identify the range of parameters for it to be an $\mathbb{F}^{Y}$ -semimartingale, which leads to $H \in (1/2,1)$ for GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with $H \in \{1/2\}\cup (3/4,1]$ proved by Cheridito (Bernoulli 7:913–934, 2001) and shows the significance of the additional parameter introduced in GFBM. We then study semimartingale asset pricing theory with the mixed GFBM, in the presence of long-range dependence, and applications in option pricing and portfolio optimisation. Finally, we discuss the implications on arbitrage theory of using GFBM, providing in particular an example of a semimartingale asset pricing model with long-range dependence without arbitrage.
