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

Author

Listed:
  • Tomoyuki Ichiba

    (University of California)

  • Guodong Pang

    (Rice University)

  • Murad S. Taqqu

    (Boston University)

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.

Suggested Citation

  • Tomoyuki Ichiba & Guodong Pang & Murad S. Taqqu, 2025. "Semimartingale properties of a generalised fractional Brownian motion and its mixtures with applications in asset pricing," Finance and Stochastics, Springer, vol. 29(3), pages 757-789, July.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:3:d:10.1007_s00780-025-00562-8
    DOI: 10.1007/s00780-025-00562-8
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    References listed on IDEAS

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    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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