Acta mathematica scientia, Series B >
CONVERGENCE RATE OF MULTIPLE FRACTIONAL STRATONOVICH TYPE INTEGRAL FOR HURST PARAMETER LESS THAN 1/2
Received date: 2010-10-06
Revised date: 2011-04-20
Online published: 2011-09-20
Supported by
The work supported by the scientific research fund of Central South University for Nationalities (YZZ09005).
In this paper, we have investigated the problem of the convergence rate of the multiple integral
{∫T0···∫T0 f(t1, ··· , tn)dBH, πt1 ···dBH, πtn},
where f ∈ Cn+1([0, T]n) is a given function, π is a partition of the interval [0, T] and {BH, πti } is a family of interpolation approximation of fractional Brownian motion BHt with Hurst parameter H < 1/2. The limit process is the multiple Stratonovich integral of the function f. In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is Δ2H in the mean square sense, where Δ is the norm of the partition generating the approximations.
WANG Bao-Bin . CONVERGENCE RATE OF MULTIPLE FRACTIONAL STRATONOVICH TYPE INTEGRAL FOR HURST PARAMETER LESS THAN 1/2[J]. Acta mathematica scientia, Series B, 2011 , 31(5) : 1694 -1708 . DOI: 10.1016/S0252-9602(11)60354-8
[1] Bardina X, Jolis M, Tudor C A. Convergence in law to the multiple fractional integral. Stoch Proc Appl, 2003, 105(2): 315–344
[2] Bardina X, Jolis M. Multiple fractional integral with Hurst parameter less than 1/2 . Stoch Proc Appl, 2006, 116: 463–479
[3] Huang S T, Cambanis S. Stochastic and multiple Wiener integrals for Gaussian processes. Ann Probab, 1978, 6(4): 585–614
[4] Hu Y. Integral transformations and anticipative calculus for fractional Brownian motions. Mem Amer Math Soc, 2005, 175(825)
[5] Hu Y, Wang B B. Convergence rate of an approximation to multiple integral of FBM. Acta Math Sci, 2010, 30B(3): 975–992
[6] Wang B B. Multiple Stratonovich Integral of Fractional Brownian Motion and its Convergence Rate[D]. Wuhan: Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, 2008 (in Chinese)
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