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

Tomoyuki Ichiba (), Guodong Pang () and Murad S. Taqqu ()
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Tomoyuki Ichiba: University of California
Guodong Pang: Rice University
Murad S. Taqqu: Boston University

Finance and Stochastics, 2025, vol. 29, issue 3, No 4, 757-789

Abstract: Abstract 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.

Keywords: Generalised fractional Brownian motion; Semimartingale; Mixture of Brownian motion and GFBM; Long-range dependence; Stock price model; Option pricing; Portfolio optimisation; Arbitrage; 60G22; 60G44; 91G30; 91G10 (search for similar items in EconPapers)
JEL-codes: C61 G11 G12 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00780-025-00562-8

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