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Approximation of Fractional Brownian Motion by Martingales

Abstract

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exists a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.

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References

  1. Androshchuk T, Mishura Y (2006) Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. (English) Stochastics 78(5):281–300

    MATH  MathSciNet  Google Scholar 

  2. Bashirov AE (2003) Partially Observable Linear Systems Under Dependent Noises. Basel: Birkhäuser

    Book  MATH  Google Scholar 

  3. Delgado R, Jolis M (2000) Weak approximation for a class of Gaussian process. J Appl Probab 37:400–407

    Article  MATH  MathSciNet  Google Scholar 

  4. Dung NT (2011) Semimartingale approximation of fractional Brownian motion and its applications. Comput Math Appl 61(7):1844–1854

    Article  MATH  MathSciNet  Google Scholar 

  5. Dzhaparidze K, van Zanten H (2004) A series expansion of fractional Brownian motion. Probab Theory Relat Fields 130(1):39–55

    Article  MATH  Google Scholar 

  6. Enriquez N (2004) A simple construction of the fractional Brownian motion. Stoch Process their Appl 109:203–223

    Article  MATH  MathSciNet  Google Scholar 

  7. Hiriart-Urruty JB, Lemaréchal C (2001) Convex analysis and minimization algorithms. In: Part 1: Fundamentals. Grundlehren der Mathematischen Wissenschaften 305. Berlin: Springer-Verlag

    Google Scholar 

  8. Li Y, Dai H (2011) Approximations of fractional Brownian motion. Bernoulli 17(4):1195–1216

    Article  MATH  MathSciNet  Google Scholar 

  9. Meyer Y, Sellan F, Taqqu MS (1999) Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J Fourier Anal Appl 5:465–494

    Article  MATH  MathSciNet  Google Scholar 

  10. Mishura YS (2008) Stochastic calculus for fractional Brownian motion and related processes. In: Lecture Notes in Mathematics 1929. Berlin: Springer

    Google Scholar 

  11. Mishura YS, Banna OL (2009) Approximation of fractional Brownian motion by Wiener integrals. Theory Probab Math Stat 79:107–116

    Article  MathSciNet  Google Scholar 

  12. Norros I, Valkeila E, Virtamo J (1999) An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4):571–587

    Article  MATH  MathSciNet  Google Scholar 

  13. Ral’chenko KV, Shevchenko GM (2010) Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations. translation in Ukr Math J 62(9):1460–1475

    MathSciNet  Google Scholar 

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Corresponding author

Correspondence to Georgiy Shevchenko.

Additional information

The second and third authors are grateful to European commission for the support within Marie Curie Actions program, grant PIRSES-GA-2008-230804.

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Shklyar, S., Shevchenko, G., Mishura, Y. et al. Approximation of Fractional Brownian Motion by Martingales. Methodol Comput Appl Probab 16, 539–560 (2014). https://doi.org/10.1007/s11009-012-9313-8

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Keywords

  • Fractional Brownian motion
  • Martingale
  • Approximation
  • Convex functional

Mathematics Subject Classifications (2010)

  • 60G22
  • 60G44
  • 90C25