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Solutions to BSDEs Driven by Multidimensional Fractional Brownian Motions
We study more general backward stochastic differential equations driven by multidimensional fractional Brownian motions. Introducing the concept of the multidimensional fractional (or quasi-) conditional expectation, we study some of its properties. Using the quasi-conditional expectation and multidimensional fractional Itô formula, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions, where a fixed point principle is employed. Finally, solutions to linear fractional backward stochastic differential equations are investigated.
Increasing attention has been paid to fractional Brownian motion recently, since it can more accurately describe many phenomena in mathematical finance, hydrology, queueing theory, and so on. A centered Gaussian process is called fractional Brownian motion of Hurst parameter if it has the covariance function . This process was first introduced by Kolmogorov  and studied by Mandelbrot and van Ness , where a stochastic integral representation in terms of a standard Brownian motion was established. Unfortunately, is neither a semimartingale nor a Markov process, so the powerful tools from the classical theories are not applicable when studying . Nevertheless, an efficient stochastic calculus of has been developed in [3–5]. This calculus uses an Itô type integration with respect to and white noise theory. The theory was extended to multiparameter fractional Brownian motion fields , , in  and applied to stochastic differential equations driven by such fractional white noise in . In many applications such as mathematical finance, the stochastic calculus of a fractional Brownian motion is needed (see [7, 8]).
Pardoux and Peng  first proved the existence and uniqueness of the solution of general nonlinear BSDEs. Since then, the theory of BSDEs has been extensively studied by many researchers and BSDEs have found applications in many fields, such as finance, risk measure, stochastic control, and so forth. These equations are of the formwhere is a standard Brownian motion and the terminal value and generator are given.
In  Hu and Peng studied the backward stochastic differential equations driven by a fractional Brownian motion. The equations are as follows:where , is a continuous function, with , , and being deterministic constants or functions, and is a fractional Brownian motion. They introduced the quasi-conditional expectation and first discussed its properties. They proved the existence and uniqueness of the solutions to above equations (2). Whenthe existence of the solutions was proved in . Zhang  studied properties of solutions of linear fractional BSDEs using the quasi-conditional expectation and derived the comparison theorem and comonotonic theorem of the solutions.
Maticiuc and Nie  improved the result and omitted the assumption in ; they proved an existence and uniqueness of the solution of reflected BSDEs driven by fractional Brownian motion. Jańczak-Borkowska  added an additional term integral with respect to an increasing process in  and proved that kind of equation has a unique solution.
Fei et al.  extended the results in  to the BSDEs driven by both standard and fractional Brownian motion:where . Using the quasi-conditional expectation, they discussed the existence and uniqueness of solutions to general BSDEs.
In this paper we will deal with the backward stochastic differential equations driven by multidimensional fractional Brownian motions with a more general terminal condition:where andwhere , , are independent fractional Brownian motions of Hurst parameter with . We use a fixed point principle to prove the existence and uniqueness of solutions to BSDEs (5).
The rest of the paper is organized as follows. In Section 2, we recall some basic results of multidimensional fractional Brownian motion. Section 3 is concerned with the multidimensional fractional (or quasi-) conditional expectation. In Section 4, existence and uniqueness of the solutions to the general fractional BSDEs are proved. Solutions of the linear fractional BSDEs are discussed in Section 5. The conclusions are set out in Section 6.
In this section, we first recall some basic results of multidimensional fractional Brownian motions.
Let , , be independent fractional Brownian motions of Hurst parameters with on a filtered probability space , where the filtration is the -algebra generated by . The process satisfieswhere
Let and be two continuous functions on . Definewhere . If , then we denoteIf is a given function, let of at time be the Malliavin -derivative; thenwhere is Malliavin derivative of at time . More detailed study about Malliavin derivative is in .
We now give the multidimensional fractional Itô formula.
Lemma 1 (multidimensional fractional Itô formula [3, 17]). For , , , let be deterministic functions and let be continuously differentiable as functions of . Denote :where is a constant vector and , , are continuous deterministic functions with . Let be continuously differentiable with respect to and twice continuously differentiable with respect to . Then
Since , we obtainwhereBy the two-dimensional fractional Itô formula, we can obtain the following chain rule.
Lemma 2 (fractional Itô chain rule [3, 17]). Let , , , , , , be real-valued stochastic processes such thatAssume also that and , , , are continuously differentiable with respect to for almost all . Suppose that and , , . DenoteThenwhich can be written as a differential form:
We know that fractional Brownian motion and Brownian motion have the following relationship:where is a standard Brownian motion andwith being a constant such that . With this we associate an operatorThis operator has an inverse operator which can be expressed aswhere (see )Let us consider the translationwhere is a continuous and adapted process; thenwhereBy the Novikov condition, ifthenwhich is an exponential martingale. Define by the following:Then, under the probability measure , is a new fractional Brownian motion.
3. Quasi-Conditional Expectation
In this section, we introduce multidimensional fractional (or quasi-) conditional expectation.
Let . denotes the completion space of continuous functions under the Hilbert norm . Let denote the set of all real symmetric functions of variables on such thatLet denote the set of such that has the following chaos expansion:where .
Now we introduce the quasi-conditional expectation for multidimensional fractional Brownian motion, which is important in solving backward stochastic differential equations.
Definition 3. The quasi-conditional expectation of some random variable relative to a fractional Brownian motion with Hurst parameter is defined bywhere .
Let , since , , are independent fractional Brownian motions; by Definition 3, we have
Lemma 4. If is the real stochastic processes such thatand , then
We introduce a function
Let be a standard Brownian motion started at and let be a -matrix; one defines an -valued Markov process by stipulation that if a.s., then , where is -dimensional Brownian motion. If is continuous, then we havewhere , with being the transpose of (see ).
Theorem 5. Let , , , and , where . For , , and , assume that , , , are measurable and . Let be -matrix, whereIf almost surely, thenWe also have
Proof. It suffices to show the case . Applying Lemma 1 to , we haveBy (38) and the chain rule of composite function, we deduceBy (42) and (43), we obtainTaking the quasi-conditional expectation both sides of (44), we haveButfrom the semigroup property of ; we haveand it is easy to check thatHenceThe assertion is completed.
4. Backward Stochastic Differential Equations Driven by Multidimensional Fractional Brownian Motions
In this section we will consider a backward stochastic differential equation driven by multidimensional fractional Brownian motions.
Denote , whereand are independent fractional Brownian motions of Hurst parameters .
Assume that, , are given constants.For , are continuous deterministic functions.For , , are continuous deterministic functions, and , where .For , ,where .
Now we consider the following backward stochastic differential equation:That is,where .
We further assume that is a continuously differentiable function with respect to and satisfies polynomial growth. is a continuous function with respect to and twice continuously differentiable with respect to , , and , so there exists a constant , such that, for all , , , , and , , we have
The partial differential equation associated with above equation (52) is
Theorem 6. If (55) has a solution which is continuously differentiable with respect to and twice continuously differentiable with respect to , then
Lemma 7. Let , , be continuous with respect to and continuously differentiable with respect to and let them be of polynomial growth. Assume that , ifthen
Theorem 8. Assume that and let (52) have a solution of the form . Then
In order to show the existence and uniqueness of solution to (52), we introduce the following set.
; is continuously differentiable with respect to and twice continuously differentiable with respect to .
And let denote the set of the process of the form
Let be the completion of under the following -norm:where is a positive arbitrary real-valued parameter.
We want to show that the solution to (52) in exists uniquely. Let , be given. Consider the following backward stochastic differential equation:
Theorem 9. Assume that and let and ; then the solution to (69) satisfies .
Proof. By (69), we knowTaking the quasi-conditional expectation, we havewhere . Obviously, we have being in .
By (70), let ; we getTaking the quasi-conditional expectation, we deduceBy the fractional Clark formula , we havewhereFrom the expressions of , , we easily infer that , . We complete the proof.
Now we give the theorem of existence and uniqueness of the solution to BSDE (52).
Theorem 10. Assume that ; then BSDE (52) has a unique solution in .
Proof. Let , , , . From Theorem 8, we know that , , , , satisfy (70) with possibly different terminal conditions:DenoteThen we getand we deduceHowever, if , by Theorem 8, we haveBy the property of Malliavin derivative relative to the fractional Brownian motion, we obtain