Advances in Mathematical Physics

Volume 2013 (2013), Article ID 827192, 11 pages

http://dx.doi.org/10.1155/2013/827192

Research Article

The -Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs

Zhi Wang1,2 and Litan Yan³

¹College of Information Science and Technology, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China

²School of Sciences, Ningbo University of Technology, 201 Fenghua Road, Ningbo 315211, China

³Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China

Received 28 May 2013; Revised 7 June 2013; Accepted 8 June 2013

Academic Editor: Ming Li

Copyright © 2013 Zhi Wang and Litan Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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. [3] and Nualart [4]). 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. [5], Liu and Yan [6], Shen and Chen [7], 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 [15], 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 [16]. Recently, motivated by stochastic control problems, Biagini et al. [17] first studied linear BSDEs driven by a fractional Brownian motion, where existence and uniqueness were discussed in order to study a maximum principle. Bender [18] gave explicit solutions for a linear BSDEs driven by a fractional Brownian motion, and Hu and Peng [19] 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 [20], Geiss et al. [21], Karoui et al. [22], Ma et al. [23], Maticiuc and Nie [24], Peng [25], 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 [26], Biagini et al. [27], Hu [28], Mishura [29], Li [30], Li and Zhao [31, 32], and Lim and Muniandy [33]). 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. [1] 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.

2. Preliminaries

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. [3], Nualart [4] and Tudor [9, 10], Zähle [34], 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 [35]). 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 [35]). 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 any one 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.

Remark 8. Under the assumptions of Theorem 7, exists as a Pettis integral (see Definition 2.3 in [35]). In fact, for all ,

Thus, the property of the Pettis integral deduces

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 [36]. 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 [35], 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 [18]). 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 [18], suppose that , which fulfills the conditions of Theorem 13 for all and . Consequently,

Next, according to Definition 4 and the growth condition, exists when tends to zero.

On the other hand, similar to (58), we obtain

By the growth and the continuity conditions of , we have converges to as tends to zero.

Now it remains to show the existence of the last integral of (72). In fact, there exists a constant , such that

For , (48) and yield

For , , we obtain

This means that is well defined, which completes the proof.

The above theorem also holds for geometric sub-fBm as described in the following proposition.

Proposition 18. Let a geometric sub-fBm , and is continuous and of polynomial growth. Then Theorem 17 holds with the terminal value of the form .

Proof. We just need to apply Theorem 17 with and .

The regularity of the obtained solutions is described as follows.

Proposition 19. Let , as defined in Theorem 17. Then . Moreover, when .

It is a straightforward result in view of the growth condition of .

6. Conclusion

We have presented the subfractional Itô integral using the method of the -transform. A Girsanov theorem with respect to the subfractional Itô integral and an Itô formula for functionals of a subfractional Wiener integral has been established. As an application, we obtain explicit solutions for a class of linear BSDEs driven by a sub-fBm with arbitrary Hurst parameter under suitable assumptions.

Acknowledgments

The Project was sponsored by NSFC (11171062, 40901241), Innovation Program of Shanghai Municipal Education Commission(12ZZ063), the Research Project of Education of Zhejiang Province (Y201326507) and Natural Science Foundation of Zhejiang Province (Y5090377).

References

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, “Sub-fractional Brownian motion and its relation to occupation times,” Statistics & Probability Letters, vol. 69, no. 4, pp. 405–419, 2004.

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, “Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems,” Electronic Communications in Probability, vol. 12, pp. 161–172, 2007.

E. Alòs, O. Mazet, and D. Nualart, “Stochastic calculus with respect to Gaussian processes,” The Annals of Probability, vol. 29, no. 2, pp. 766–801, 2001.

D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, Germany, 2nd edition, 2006.

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, “Fractional Brownian density process and its self-intersection local time of order ,” Journal of Theoretical Probability, vol. 17, no. 3, pp. 717–739, 2004.

J. Liu and L. Yan, “Remarks on asymptotic behavior of weighted quadratic variation of subfractional Brownian motion,” Journal of the Korean Statistical Society, vol. 41, no. 2, pp. 177–187, 2012.

G. Shen and C. Chen, “Stochastic integration with respect to the sub-fractional Brownian motion with ,” Statistics & Probability Letters, vol. 82, no. 2, pp. 240–251, 2012.

C. Tudor, “Some properties of the sub-fractional Brownian motion,” Stochastics, vol. 79, no. 5, pp. 431–448, 2007.

C. Tudor, “Inner product spaces of integrands associated to subfractional Brownian motion,” Statistics & Probability Letters, vol. 78, no. 14, pp. 2201–2209, 2008.

C. Tudor, “Some aspects of stochastic calculus for the sub-fractional Brownian motion,” Analele Universitatii Bucuresti. Matematica, vol. 57, no. 2, pp. 199–230, 2008.

C. Tudor, “On the Wiener integral with respect to a sub-fractional Brownian motion on an interval,” Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 456–468, 2009.

L. Yan, K. He, and C. Chen, “The generalized Bouleau-Yor identity for a sub-fractional Brownian motion,” Science China Mathematics, 2013.

L. Yan and G. Shen, “On the collision local time of sub-fractional Brownian motions,” Statistics & Probability Letters, vol. 80, no. 5-6, pp. 296–308, 2010.

L. Yan, G. Shen, and K. He, “Itô's formula for a sub-fractional Brownian motion,” Communications on Stochastic Analysis, vol. 5, no. 1, pp. 135–159, 2011.

É. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990.

S. Peng, “Backward stochastic differential equations,” in Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, Lecture Notes in Chinese Summer School in Mathematics Weihai, 2004.

F. Biagini, Y. Hu, B. Øksendal, and A. Sulem, “A stochastic maximum principle for processes driven by fractional Brownian motion,” Stochastic Processes and their Applications, vol. 100, pp. 233–253, 2002.

C. Bender, “Explicit solutions of a class of linear fractional BSDEs,” Systems & Control Letters, vol. 54, no. 7, pp. 671–680, 2005.

Y. Hu and S. Peng, “Backward stochastic differential equation driven by fractional Brownian motion,” SIAM Journal on Control and Optimization, vol. 48, no. 3, pp. 1675–1700, 2009.

J.-M. Bismut, “Conjugate convex functions in optimal stochastic control,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 384–404, 1973.

C. Geiss, S. Geiss, and E. Gobet, “Generalized fractional smoothness and -variation of BSDEs with non-Lipschitz terminal condition,” Stochastic Processes and their Applications, vol. 122, no. 5, pp. 2078–2116, 2012.

N. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.

J. Ma, P. Protter, and J. M. Yong, “Solving forward-backward stochastic differential equations explicitly—a four step scheme,” Probability Theory and Related Fields, vol. 98, no. 3, pp. 339–359, 1994.

L. Maticiuc and T. Nie, “Fractional backward stochastic differential equations and fractional backward variational inequalities,” http://arxiv.org/abs/1102.3014

S. G. Peng, “Backward stochastic differential equations and applications to optimal control,” Applied Mathematics and Optimization, vol. 27, no. 2, pp. 125–144, 1993.

B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968.

F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2006.

Y. Hu, “Integral transformations and anticipative calculus for fractional Brownian motions,” Memoirs of the American Mathematical Society, vol. 175, no. 825, 2005.

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, Germany, 2008.

M. Li, “On the long-range dependence of fractional Brownian motion,” Mathematical Problems in Engineering, vol. 2013, Article ID 842197, 5 pages, 2013.

M. Li and W. Zhao, “On noise,” Mathematical Problems in Engineering, vol. 2012, Article ID 673648, 23 pages, 2012.

M. Li and W. Zhao, “Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise,” Physica A, vol. 391, no. 24, pp. 6268–6278, 2012.

S. C. Lim and S. V. Muniandy, “On some possible generalizations of fractional Brownian motion,” Physics Letters A, vol. 266, no. 2-3, pp. 140–145, 2000.

M. Zähle, “Integration with respect to fractal functions and stochastic calculus. I,” Probability Theory and Related Fields, vol. 111, no. 3, pp. 333–374, 1998.

C. Bender, “An -transform approach to integration with respect to a fractional Brownian motion,” Bernoulli, vol. 9, no. 6, pp. 955–983, 2003.

C. Bender, “An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter,” Stochastic Processes and their Applications, vol. 104, no. 1, pp. 81–106, 2003.