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A new proof of fractional Hu-Meyer formula and its applications
Journal of Inequalities and Applications volume 2012, Article number: 272 (2012)
Abstract
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.
MSC:60G15, 62H05.
1 Introduction
It is well known that Hu and Meyer [1] 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. [2] 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 [3] and the references therein.
Bardina et al. [4] 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 [5], we define the integral of Stratonovich type in the mollifier approximation sense. Unlike our construction, in the paper [4], the Stratonovich integral is defined in the Solé-Utzet sense (see [6]). 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 [7], 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 [3] 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
In this paper, we denote by the basic probability space. The expectation on this basic probability space is denoted by . The fBm (, ) of the Hurst parameter H is a Gaussian process with mean 0 and covariance given by
Throughout this paper, we assume .
For a fixed positive integer n and a suitable (deterministic) function of n variables the multiple Itô integral
and the multiple Stratonovich integral
are well defined (see [3, 8] and the references therein).
Following the notations in [9], we define
with
This implies is obtained by using a variable y instead of x.
For a continuous function of n variables , we define
This means is obtained from f by replacing the k th variable by s.
3 Hu-Meyer formula
Now, the Wick product ⋄ of two functionals is introduced. To extend the theory of stochastic calculus from Brownian motions to fBms, the Wick calculus for Gaussian processes (or Gaussian measures) is used. The Wick product of two exponential functions (see [2]) and is defined as
where
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.
Proposition 1 Let X and Y be two random variables. Then we have
Proof By the definition of an exponential function,
using the expression (3.1)
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. □
As in [5], for and fixed t, as ε tends to zero, tends to the Dirac function at t, . Define
Obviously,
when ε tends to zero. Then is a Gaussian random variable (see [10]). Furthermore, from [4], we have
Lemma 1 For defined by (3.2), , we have
where run over all permutations of and , .
Proof Let (), by (3.1) we obtain
Then, by the definition of an exponential function,
Next we will compare the coefficients of . On the one hand, it is obvious that the left-hand side of (3.5) is equal to
therefore, the coefficient of the term on the left-hand side is .
On the other hand, the right-hand side of (3.5) coincides with
Notice that the coefficient of on the right-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)
Let . There exists the limit in of
and the limit is given by the extended Hu-Meyer formula
where
with the convention that .
Remark 1 This result is not the same as Theorem 4.4 in [4], where the traces that appear are defined by a limit procedure, not in the way stated here.
Proof Using Lemma 1 and the property of the Wick product, we have that
where
Submitting (3.3) to the above expression,
By the continuity of the multiple Itô-type integrals on the spaces [11], it follows that
which is in the sense as .
Denote
where
It is easy to prove that converge to in the same way as in [4]. Since σ are the permutations of , we get the desired result. □
4 Applications to the convergence rate of the multiple Stratonovich integral
To complement the paper, we introduce some notations. Let be a partition of the interval . Denote
Without ambiguity, we will also denote the interval by . We also consider a class of partitions Π such that
Let be the interpolation approximation of ,
where
Consider the approximation of the multiple stochastic integral
It is proved in [4] 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.
Theorem 2 Suppose that . Given a sequence partition π of the interval satisfying (4.1), there is a random variable such that converges to in the mean square sense. Moreover, there is a constant C, independent of partition π, such that
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.
Proposition 2 Assume , and , . Let f continuously bound first and second derivatives on . Then we have
Proof Notice that
where
Since , , we get
Similarly, we also have .
We denote , , a simple computation implies that
and
where is a bounded constant.
Denote
obviously, . According to (4.4) and (4.5), we have
the proof is complete. □
It is easy to obtain the following result by calculation.
Lemma 2 Let . If we denote
then we have
Lemma 3 Suppose that . If we denote
we have
and
Proof It follows easily by induction. □
Lemma 4 Suppose that . Then, for , , ,
Proof Observe that for , , ,
where
It is obvious that .
Set
where
Moreover, .
Without loss of generality, we can write
Clearly, A is a bounded constant.
Putting together (4.6) and (4.7) and using Lemma 3, we get that
□
Proof of Theorem 2 If we take , then we have the polygonal approximation (4.2). By using Theorem 1, it is easy to see that
Set
Then
Using the properties of multiple Wiener-Itô integrals (see [11]), we derive the following:
In order to prove (4.3), we will check for all .
For , we get
For , we also write
where .
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.
On the one hand, by Lemma 2 and Lemma 3, we obtain
Note that
On the other hand, clearly, we have
Therefore, we obtain
and the proof is complete. □
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Acknowledgements
The work described in this paper is supported by National Natural Science Foundation of China under Grants [No. 11201348].
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BW proposed the problem and finished the proof. TH gave BW some useful advice to improve the convergence rate. All authors read and approved the final manuscript.
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Wang, B., Hu, T. A new proof of fractional Hu-Meyer formula and its applications. J Inequal Appl 2012, 272 (2012). https://doi.org/10.1186/1029-242X-2012-272
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DOI: https://doi.org/10.1186/1029-242X-2012-272