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Bifractional Brownian motion for H>1 and 2HK≤1

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  • Talarczyk, Anna

Abstract

Bifractional Brownian motion on R+ is a two parameter centered Gaussian process with covariance function: RH,K(t,s)=12Kt2H+s2HK−|t−s|2HK,s,t≥0.This process has been originally introduced by Houdré and Villa in Houdré and Villa (2002) in for the range of parameters H∈(0,1] and K∈(0,1]. Since then, the range of parameters, for which RH,K is known to be nonnegative definite has been somewhat extended, but the full range is still not known. We give an elementary proof that RH,K is nonnegative definite for parameters H,K satisfying H>1 and 0<2HK≤1. We show that RH,K can be decomposed into a sum of two nonnegative definite functions. As a side product we obtain a decomposition of the fractional Brownian motion with Hurst parameter H<12 into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. We also discuss some simple properties of bifractional Brownian motion with H>1.

Suggested Citation

  • Talarczyk, Anna, 2020. "Bifractional Brownian motion for H>1 and 2HK≤1," Statistics & Probability Letters, Elsevier, vol. 157(C).
    • Handle: RePEc:eee:stapro:v:157:y:2020:i:c:s0167715219302743
    DOI: 10.1016/j.spl.2019.108628
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    Cited by:

    1. Chunsheng Ma, 2023. "Vector Random Fields on the Probability Simplex with Metric-Dependent Covariance Matrix Functions," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1922-1938, September.

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