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Journals / Hacettepe Journal of Mathematics and Statistics / 2011 / Cilt: 40 - Sayı: 2

Results on the supremum of fractional Brownian motion

Ceren VARDAR
Pages
255–264
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

Finally, applications of the given results to financial markets are investigated, and partial results are provided.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We show that the distribution of the square of the supremum of reflected fractional Brownian motion up to time a, with Hurst parameter H greater than 1/2, is related to the distribution of its hitting time to level 1, using the self similarity property of fractional Brownian motion.
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Key Findings & Data:It is also proven that the second moment of supremum of reflected fractional Brownian motion up to time a is bounded above by $a^ {2H}$.
💡
Conclusion & Impact:Finally, applications of the given results to financial markets are investigated, and partial results are provided.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

We show that the distribution of the square of the supremum of reflected fractional Brownian motion up to time a, with Hurst parameter H greater than 1/2, is related to the distribution of its hitting time to level 1, using the self similarity property of fractional Brownian motion. It is also proven that the second moment of supremum of reflected fractional Brownian motion up to time a is bounded above by $a^ {2H}$. Similar relations are obtained for the supremum of fractional Brownian motion with Hurst parameter greater than 1/2, and its hitting time to level 1. What is more, we obtain an upper bound on the complementary probability distribution of the supremum of fractional Brownian motion and re°ected fractional Brownian motion up to time a, using Jensen's and Markov's inequalities. A sharper bound is observed on the distribution of the supremum of fractional Brownian motion by the properties of Gamma distribution. Finally, applications of the given results to financial markets are investigated, and partial results are provided.

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Cite This Article

VARDAR, C. (2022). Results on the supremum of fractional Brownian motion. Hacettepe Journal of Mathematics and Statistics, Cilt: 40 - Sayı: 2, 255–264. https://ajindex.com/articles/530537

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References (13)

  1. Abry, P. and Sellan, F. The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation, Appl. Comp. Harmonic Anal. 3, 337{383, 1996
  2. Bertoin, J. Levy Processes (Cambridge University Press, Cambridge, UK, 1996)
  3. Biagini, F., Hu, Yaozhong., Oksendal, B. and Zhang, T. Stochastic Calculus for Fractional Brownian Motion and Applications (Springer Verlag, London, 2008)
  4. Caglar, M. Simulation of fractional Brownian motion with micropulses, Advances in Per- formance Analysis 3 (1), 43{69, 2000
  5. Decreusefond, L. and Nualart, D. Hitting times For Gaussian processes, Annals Probab. 36, 319{330, 2008
  6. Douady, R., Yor, M. and Shiryaev, A. N. On the probability charateristics of drop" variables in standard Brownian motion, Theory Probab. Appl. 44 (1), 29{38, 2000
  7. Kolmogorov, A. N. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. URSS(N.S) 26, 115{118, 1940
  8. L¶evy, P. Processus Stochatiques et Mouvement Brownien (Gauthier-Villars, Paris, 1948)
  9. Mandelbrot, B. B. and Van Ness, J. W. Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10, 422{437, 1968
  10. Revuz, D. and Yor, M. Continuous Martingales and Brownian Motion (Springer Verlag, Berlin, Hiedelberg, 2001)
  11. Rogers, L. C. G. Arbitrage with fractional Brownian motion, Mathematical Finance 7, 95{ 105, 1997
  12. Salminen, P. and Vallois, P. On maximum increase and decrease of Brownian motion, An- nales de l'institut Henri Poincare (B) Probabilit¶Us et Statistiques 43 (6), 655{676, 2007
  13. Sellan, F. Synth¶ese de mouvements Browniens fractionaires µa l'aide de la transformation par ondelettes, C. R. Acad. Sci. Paris Ser. Math. 321, 351{358, 1995.