Semimartingale properties of a generalised fractional Brownian motion and its mixtures with applications in asset pricing

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 $X$ is self-similar and has non-stationary increments, whose Hurst index H ∈ ( 0 , 1 ) $H \in (0,1)$ 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 F X $\mathbb{F}^{X}$ . We next study the mixed process Y $Y$ made up of an independent BM and a GFBM and identify the range of parameters for it to be an F Y $\mathbb{F}^{Y}$ -semimartingale, which leads to H ∈ ( 1 / 2 , 1 ) $H \in (1/2,1)$ for GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with H ∈ { 1 / 2 } ∪ ( 3 / 4 , 1 ] $H \in \{1/2\}\cup (3/4,1]$ 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.