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Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

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  • El-Beltagy, Mohamed
  • Etman, Ahmed
  • Maged, Sroor

Abstract

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.

Suggested Citation

  • El-Beltagy, Mohamed & Etman, Ahmed & Maged, Sroor, 2022. "Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    • Handle: RePEc:eee:chsofr:v:156:y:2022:i:c:s0960077922000583
    DOI: 10.1016/j.chaos.2022.111847
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    References listed on IDEAS

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    1. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    2. Cipian Necula, 2008. "Barrier Options and a Reflection Principle of the Fractional Brownian Motion," Advances in Economic and Financial Research - DOFIN Working Paper Series 6, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
    3. Cipian Necula, 2008. "Option Pricing in a Fractional Brownian Motion Environment," Advances in Economic and Financial Research - DOFIN Working Paper Series 2, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
    4. Burgos, C. & Cortés, J.-C. & Villafuerte, L. & Villanueva, R.-J., 2017. "Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 305-318.
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