UC Santa Barbara

We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial asset pricing. The GFBM is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the region of these two parameter values where the GFBM is a semimartingale. Specifically, in one region resulting in $H\in (1/2,1)$, it is in fact a process of finite variation and differentiable, and in another region also resulting in $H\in (1/2,1)$ it is not a semimartingale. For regions resulting in $H \in (0,1/2]$ except the Brownian motion case, the GFBM is also not a semimartingale. We establish $p$-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when $H < 1/2$. We next show that the mixed process made up of an independent BM and a GFBM is a semimartingale when the Hurst parameter for the GFBM is $H \in (1/2,1)$ in both regions mentioned above, and 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 (2001) and shows the significance of the additional parameter introduced in the GFBM. We then study the semimartingale asset pricing theory with the mixed GFBM, in presence of long range dependence. Our work extends Cheridito (2001) on the mixed FBM asset pricing model in which the Hurst parameter of the FBM is $H \in (3/4,1)$, because the Hurst parameter range of the GFBM is enlarged to $(1/2,1)$. In addition, when the GFBM is a process of finite variation mentioned above (resulting in $H\in (1/2,1)$), the mixed GFBM process as a stock price model is a Brownian motion with a random drift of finite variation.