A new proof of fractional Hu-Meyer formula and its applications
© Wang and Hu; licensee Springer 2012
Received: 20 March 2012
Accepted: 1 November 2012
Published: 27 November 2012
This paper is concerned with the Hu-Meyer formula for fractional Brownian motion with the Hurst parameter less than . By the mollifier approximation, the Hu-Meyer formula is explicitly obtained based on the multiple Stratonovich integral, and the proof is different from the known methods. Moreover, the obtained Hu-Meyer formula can be applied to derive the convergence rate of the multiple fractional Stratonovich integral.
It is well known that Hu and Meyer  introduced a new multiple stochastic integral with respect to a Wiener process, called a multiple Stratonovich integral, which is in general different from the usually studied multiple Wiener-Itô integral. The authors also proposed a formula (called Hu-Meyer formula) that gives the relationship of the Stratonovich integral with the Itô integrals of some functions called the traces that involve integrals of f on the diagonals.
An increasing interest is visible in the last decade in modeling long dependence phenomena in the fields of dynamical system, economics, hydrology, telecommunication network by using fractional Brownian motion (fBm for short). The fBm is a suitable generalization of standard Brownian motion which exhibits long-range dependence.
Recently, many authors have considered an integral with respect to fBm. Duncan et al.  employed the Wick products to define a fractional stochastic integral whose mean is zero. This property is very convenient for both theoretical development and practical applications. For more details, one can see  and the references therein.
Bardina et al.  constructed a multiple Stratonovich integral with respect to fBm with the Hurst parameter under some conditions. They defined the traces to obtain the Hu-Meyer formula that gives the Stratonovich integral as a sum of Itô integrals of these traces.
In this paper, we consider a similar problem for the multiple Stratonovich integral. Inspired by , we define the integral of Stratonovich type in the mollifier approximation sense. Unlike our construction, in the paper , the Stratonovich integral is defined in the Solé-Utzet sense (see ). Our aim here is to present a new proof of the Hu-Meyer formula for fBm. We also do not make use of the integral representation of fBm in terms of ordinary Brownian motion as in , where the hypothesis involves the transferring operator which is difficult to verify.
We have organized the paper as follows. Section 2 recalls some results from  on the multiple Stratonovich integral, which will be used in the remainder of the paper. Section 3 gives the Hu-Meyer formula and its proof. As an application, the fourth section is devoted to the convergence rate of the multiple fractional Stratonovich integral.
2 Multiple Stratonovich integral
Throughout this paper, we assume .
This implies is obtained by using a variable y instead of x.
This means is obtained from f by replacing the k th variable by s.
3 Hu-Meyer formula
Using the linear property, we can generalize the Wick product to the linear combination of exponential functionals. Then the Wick product can be extended to a general random variable by taking limit.
we will compare the coefficients of the term in the two sides of the above equality. Observe that the coefficient of in the left is and the one in the right is . This fact implies the truth of the proposition. □
where run over all permutations of and , .
therefore, the coefficient of the term on the left-hand side is .
where σ are the permutations of . This completes the proof. □
Let us state the main result of this section. The following explains the relations between the multiple Itô integral and the Stratonovich integral.
Theorem 1 (Hu-Meyer formula)
with the convention that .
Remark 1 This result is not the same as Theorem 4.4 in , where the traces that appear are defined by a limit procedure, not in the way stated here.
which is in the sense as .
It is easy to prove that converge to in the same way as in . Since σ are the permutations of , we get the desired result. □
4 Applications to the convergence rate of the multiple Stratonovich integral
It is proved in  that, under some mild conditions, converges to in the mean square sense. Then the natural question is: what is the precise asymptotic, i.e., convergence rate?
Our main result in this section is stated as follows.
We must point out that the Hu-Meyer formula will be the key tool used in order to obtain the convergence rate of the interpolation approximation for general n considered in the section. In order to prove the above theorem, we also need the following results.
Similarly, we also have .
where is a bounded constant.
the proof is complete. □
It is easy to obtain the following result by calculation.
Proof It follows easily by induction. □
It is obvious that .
Clearly, A is a bounded constant.
In order to prove (4.3), we will check for all .
By some elementary calculations, we know that the main terms which determine the convergence rate are and , whose expressions are similar to the correspondence terms and respectively.
and the proof is complete. □
The work described in this paper is supported by National Natural Science Foundation of China under Grants [No. 11201348].
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