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The application of fractional derivatives in stochastic models driven by fractional Brownian motion

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  • Longjin, Lv
  • Ren, Fu-Yao
  • Qiu, Wei-Yuan

Abstract

In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.

Suggested Citation

  • Longjin, Lv & Ren, Fu-Yao & Qiu, Wei-Yuan, 2010. "The application of fractional derivatives in stochastic models driven by fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4809-4818.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:21:p:4809-4818
    DOI: 10.1016/j.physa.2010.06.016
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    References listed on IDEAS

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    4. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
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    5. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xili & Zhang, Xiaoli, 2012. "Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6418-6431.
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    9. Zhao, Ailiang & Li, Junmin & Lei, Yanfang & He, Chao, 2021. "Robust point control for a class of fractional-order reaction–diffusion systems via non-collocated point measurement," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 584(C).
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