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Zhi Wang, Litan Yan, "The -Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs", Advances in Mathematical Physics, vol. 2013, Article ID 827192, 11 pages, 2013. https://doi.org/10.1155/2013/827192
The -Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs
Let be a subfractional Brownian motion with index . Based on the -transform in white noise analysis we study the stochastic integral with respect to , and we also prove a Girsanov theorem and derive an Itô formula. As an application we study the solutions of backward stochastic differential equations driven by of the form , where the stochastic integral used in the above equation is Pettis integral. We obtain the explicit solutions of this class of equations under suitable assumptions.
As an extension of Brownian motion, Bojdecki et al. [1, 2] introduced and studied a rather special class of self-similar Gaussian processes which preserves many properties of the fractional Brownian motion of the Weyl type here and below. This process arises from occupation time fluctuations of branching particle systems with Poisson initial condition. This process is called the subfractional Brownian motion (sub-fBm). The so-called sub-fBm with index is a mean zero Gaussian process with and the covariance for all , . For , coincides with the standard Brownian motion . is neither a semimartingale nor a Markov process unless . So many of the powerful techniques from stochastic analysis are not available when dealing with . As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to (see, e.g., Alòs et al.  and Nualart ). The sub-fBm has properties analogous to those of fractional Brownian motion and satisfies the following estimates: Thus, Kolmogorov's continuity criterion implies that subfractional Brownian motion is Hölder continuous of order for any . But its increments are not stationary. More works for sub-fBm can be found in Bojdecki et al. , Liu and Yan , Shen and Chen , Tudor [8–11], Yan et al. [12–14], and the references therein.
On the other hand, it is well known that general backward stochastic differential equations (BSDEs) driven by a Brownian motion were first studied by Pardoux and Peng , where they also gave a probabilistic interpretation for the viscosity solution of semilinear partial differential equations. Because of their important value in various areas including probability theory, finance, and control, BSDEs have been subject to the attention and interest of researchers. A survey and complete literature for BSDEs could be found in Peng . Recently, motivated by stochastic control problems, Biagini et al.  first studied linear BSDEs driven by a fractional Brownian motion, where existence and uniqueness were discussed in order to study a maximum principle. Bender  gave explicit solutions for a linear BSDEs driven by a fractional Brownian motion, and Hu and Peng  studied the linear and nonlinear BSDEs driven by a fractional Brownian motion using the quasi-conditional expectation. More works for the BSDEs driven by Brownian motion and fractional Brownian motion can be found in Bisumt , Geiss et al. , Karoui et al. , Ma et al. , Maticiuc and Nie , Peng , and the references therein. In this paper, we study the BSDEs driven by a sub-fBm of the form where the stochastic integral used in above equation is Pettis integral.
In recent years, there has been considerable interest in studying fractional Brownian motion due to its applications in various scientific areas including telecommunications, turbulence, image processing, and finance and also due to some of its compact properties such as long-range dependence, self-similarity, stationary increments, and Hölder's continuity (see, e.g., Mandelbrot and van Ness , Biagini et al. , Hu , Mishura , Li , Li and Zhao [31, 32], and Lim and Muniandy ). Moreover, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about self-similar Gaussian processes and fields in general. Therefore, other generalizations of Brownian motion have been introduced such as sub-fBm, bifractional Brownian motion, and weighted-fractional Brownian motion. However, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-similar Gaussian processes. The main reason for this is the complexity of dependence structures for self-similar Gaussian processes which do not have stationary increments. The sub-fBm has properties analogous to those of fractional Brownian motion (self-similarity, long-range dependence, Hölder paths, the variation, and the renormalized variation). However, in comparison with fractional Brownian motion, the sub-fBm has nonstationary increments and the increments over nonoverlapping intervals are more either weakly or strongly correlated and their covariance decays polynomially as a higher rate in comparison with fractional Brownian motion (for this reason in Bojdecki et al.  is called subfractional Brownian motion). The above mentioned properties make sub-fBm a possible candidate for models which involve long-range dependence, self-similarity, and nonstationary. Thus, it seems interesting to study the BSDEs driven by a sub-fBm.
This paper is organized as follows. Section 2 contains some basic results. In Section 3, we give a definition of subfractional Itô integral based on an -transform in white noise analysis. As an application we establish a Girsanov theorem for this integral. In Section 4, we give an Itô formula for functionals of a Wiener integral for a sub-fBm. We also discuss the geometric sub-fBm in this section. Section 5 considers the BSDEs (3). Finally, we will conclude the paper in Section 6.
In this section, we briefly recall some basic definitions and results of sub-fBm. Throughout this paper we assume that is arbitrary but fixed and let be a one-dimensional sub-fBm with Hurst index defined on . To simplify, we denote , and let be a two-sides Brownian motion and
We also denote (i): the usual -norm, and the corresponding inner product is denoted by ; (ii): the Schwartz space of rapidly decreasing smooth functions of real valued; (iii): the Wiener integral of the function ; (iv): the -field generated by ; (v): the -norm.
can be written as a Volterra process with the following moving average representation: where , , . The sub-fBm is also possible to construct a stochastic calculus of variations with respect to the Gaussian process , which will be related to the Malliavin calculus. Some surveys and complete literatures for Malliavin calculus of Gaussian process could be found in Alòs et al. , Nualart  and Tudor [9, 10], Zähle , and the references therein.
Let . Consider Weyl's type fractional integrals of order if the integrals exist for almost all , and Marchand's type fractional derivatives of order if the limit exists in for some , where for . Define the operator where and denotes the gamma function defined by Recall that we now give a stochastic version of the Hardy-Littlewood theorem as follows.
Theorem 1 (Theorem 2.10 in ). Let and let the operators be defined as above. Then is a continuous operator from into if and .
Define the function for any Borel function on . Then the function is odd, which is called the odd extension of . Based on the moving average representation (5), we can show the following proposition.
Proposition 2. Let the operators be defined as above. Then and admits the following integral representation: for all .
We finally recall the -transform. The -transform is an important tool in white noise analysis. Here we give a definition and state some results that do not depend on properties of the white noise space. Denote the -transform of (see, e.g., [35, 36] for more details) by where the Wick exponential: : of is given by The -transform has the following important properties. The -transform is injective; that is, for all , implies that . Let be a sequence that converges to in ; then: : converges to: : in . for . Hence it can deduce a probability measure on by especially, for , we can rewrite the -transform as Let be a progressively measurable process such that
Then is the unique element in with -transform given by The Wiener integral is the unique element in with -transform given by
The following result points out that the operators interchanges with the -transform.
Lemma 3 (Lemma 2.9 in ). Let exist for some . Then one has for all . In the case the convergence of the fractional derivative on the right-hand side is in the sense, if . In particular, the operators interchange with the -transform.
3. A Subfractional Itô Integral
In this section, based on the -transform we aim to define the subfractional Itô integral, denoted by with , and introduce the Girsanov theorem. To this end, inspired by the Hitsuda-Skorohod integral, we define the subfractional Itô integral as the unique random variable such that for all , provided the integral exists under suitable conditions. According to (12) and Property (), we have Combining this with the fact () in Section 2, we give the following definition.
Definition 4. Let be a Borel set. A mapping is said to be subfractional Itô integrable on if for any , and there is a such that for all .
It is important to note that in the above definition is unique because the -transform is injective, which is called the subfractional Itô integral of on and we denote it by
In this paper, sub-fractional Itô integralalways refers to the -transform approach proposed in Definition 4.
Proposition 5. The following statements hold. (1)For anyone has(2)Let be subfractional Itô integrable for . Then
Proof. These results are some simple examples.
Recall that the Wick product of is an element such that for all . The following theorem expresses the relationship between the subfractional Itô integral defined as above and the integral based on Wick product .
Theorem 6. Let and ; then in the sense that if one side is well defined then so is the other, and both coincide.
We can obtain it by calculating the -transform of both sides. In particular, for , this theorem implies that It means that the subfractional Itô integral is the -limit of Wick-Riemann sums for some suitable processes. That is, for some suitable processes , where is a partition of with and the convergence is in .
Now we calculate the expectation of a subfractional Itô integral under a measure .
Theorem 7. Let and be given by (15). If the following assumptions hold: (1) is subfractional Itô integrable, and ; (2) and for ,
One then has
Proof. Let be given such that converges to in , we have the identity
It can be easily obtained that the left-hand side of (33) converges to the same side of (32) by Theorem 1 and () in Section 2.
Then we just need to prove the right-hand side of (33) converges to (32) correspondingly. By Lemma 3, applying the fractional integration by parts rule, we have which is bounded by We can easily show that converge to zero, as , respectively, by Hölder's inequality. This completes the proof.
Now, we establish a Girsanov theorem for subfractional Itô integral. Consider the measure , , the probability space carries a two-side Brownian motion given by according to the classical Girsanov theorem. On this probability space, we denote the -transform with respect to the measure ; that is, and the following identity holds: for all .
Theorem 9. Let the assumptions of Theorem 7 be satisfied, and Then, the identity holds in -almost surely.
Proof. We apply Theorem 7 to , . It is easy to check that according to Lemma 2.5 in . By Theorem 7 and (40), it follows The second identity based on the fact that exists as a Pettis integral which is proved in Remark 8. The proof is complete.
4. An Itô Formula
In this section, we prove an Itô formula for a subfractional Wiener integral using the -transform approach. An indefinite subfractional Wiener integral is understood as a process for all provided is a deterministic function such that the above integral exists as a subfractional Itô integral for all .
Proposition 10. Assume that is continuous for , and -Hölder continuous with for . Then the indefinite subfractional Wiener integral exists, and
Proof. We should prove that and exists.
For , since is continuous on , by Hardy-Littlwood theorem, it is obvious that .
For , similar to the argument in Proposition 5.1 in , there exists a function , such that Hence, , and so is . is a deterministic function implies that exists.
Next, consider the -transform of the right-hand side in (45), then by (19), we obtain that This completes the proof.
The following lemma is essential to the proof of our Itô's formula.
Lemma 11. Let be continuous and . Then one has
In particular,(1)for all , ; (2) is differentiable in , and for all , one has
Proof. For , the following identity holds: Then, Equation (48) easily follows and the other assertions are trivial.
Remark 12. Since the right of (48) is not hold when , there is a lack of a result similar to the above Lemma. Hence, we only consider the case of constant , and we have .
Now we give the following Itô formula.
Theorem 13. Let , such that () be an indefinite subfractional Wiener integral; that is, for all , , where is continuous when , constant when ; (); () there exists constants and such that
Then the following equality holds in :
Proof. It suffices to show that both sides have the same -transform. Indeed, by Definition 4, the integral of the left-hand side has the -transform given by
Henceforth, we just need to show the right-hand side has the same result. Firstly, we show the integrals of the right-hand side exist in . Without loss of generality, denote , , , , and . By the growth condition (52), we obtain
Consequently, exists. For the last one, by Lemma 11 and Remark 12, we have
Hence, the last integral exists as a Pettis integral in the -sense.
On the other hand, denote the heat kernel as follows: Thanks to the classical Girsanov theorem, for arbitrary , under the measure , we can easily calculate that is a Gaussian random variable with mean and variance . Thus, we obtain Moreover, by , integration and differentiation can be interchanged. Since the heat kernel fulfills , we have Consequently, Compared with (54), the proof can be completed.
The objective of this part is to define the geometric sub-fBm and establish an Itô formula with respect to it.
Definition 14. Let , , and , Then one calls a geometric sub-fBm with coefficients , , , , provided the right-hand side exists as an element of for all .
Theorem 15. Let , such that (i) be a geometric sub-fBm with continuous coefficients , and let be a constant when ; (ii), hold.
Then the following equality holds in :
Proof. Let Then, apply Theorem 13 to , and the result is obvious.
The special case yields the following.
Corollary 16. Let be a geometric sub-fBm as in Theorem 15; then for all , For this reason, one calls it “geometric sub-fBm”.
5. Explicit Solution of a Class of Linear Subfractional BSDEs
General BSDEs driven by a Brownian motion are usually of the form where are given. The generator is a -adapted process for every pair , the terminal value is a -measureable random variable, and denotes the filtration generated by . We say a pair is a solution of this equation, if the processes which are -adapted and satisfy a suitable integrability condition solve the equation -almost surely.
After these preparations, we now turn to the problems to solve the BSDEs driven by a sub-fBm of the form where are given. The generator is a -adapted process for every pair , the terminal value is a -measureable random variable, and denotes the filtration generated by . We say a pair is a solution of this equation, if the processes which are -adapted and satisfy a suitable integrability condition solve the equation -almost surely.
Let us recall a result about the following PDE, which is a parabolic partial differential equation solved by the heat equation (see Theorem 9 in ). Let the following conditions be satisfied: () and is strictly increasing with and ; (); () and there exists constant and such that for all , .
Then the PDE has a classical solution given by Next we give the main result of this paper.
Theorem 17. Let and . Suppose the following conditions are satisfied: () is continuous when , constant when , and there exist constants , such that ; (); () holds with and with bounded on ; () and there exist constants , and such that for all , .
Then the BSDEs, have a solution of the form
Proof. Let ; from Lemma 11 and Remark 12, we have satisfies . By the growth condition , is yielded, and follows from . Henceforth, is a classical solution of the PDE Moreover, by Lemma 10 and Corollary 11 in , suppose that , which fulfills the conditions of Theorem 13 for all and . Consequently,