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Bounds on the expected value of maximum loss of fractional Brownian motion

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  • Vardar-Acar, Ceren
  • Bulut, Hatice

Abstract

It has been theoretically proven through present study that the expected value of maximum loss of fractional Brownian motion up to fixed time t with Hurst parameter [1/2,1) is bounded above by tHπ2 and below by tH2. These new bounds provide improvement on those bounds which have been previously derived in the literature. In order to search for closer bounds, numerical study is also performed through discretization method and multivariate Gaussian variables have been examined. The simulated values of the expected value of maximum loss of fractional Brownian motion have been provided through the use of Cholesky decomposition. As a consequence of the simulation study, it has been observed that as the Hurst parameter increases, the values of the expected maximum loss of fractional Brownian motion decreases.

Suggested Citation

  • Vardar-Acar, Ceren & Bulut, Hatice, 2015. "Bounds on the expected value of maximum loss of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 117-122.
    • Handle: RePEc:eee:stapro:v:104:y:2015:i:c:p:117-122
    DOI: 10.1016/j.spl.2015.05.001
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    References listed on IDEAS

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    1. Andrew M. Ross, 2010. "Computing Bounds on the Expected Maximum of Correlated Normal Variables," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 111-138, March.
    2. Caglar, Mine & Vardar-Acar, Ceren, 2013. "Distribution of maximum loss of fractional Brownian motion with drift," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2729-2734.
    3. Charles E. Clark, 1961. "The Greatest of a Finite Set of Random Variables," Operations Research, INFORMS, vol. 9(2), pages 145-162, April.
    4. Per A. Brodtkorb, 2006. "Evaluating Nearly Singular Multinormal Expectations with Application to Wave Distributions," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 65-91, March.
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