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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 423101, 11 pages
Classical Solutions of Path-Dependent PDEs and Functional Forward-Backward Stochastic Systems
1Institute for Financial Studies and Institute of Mathematics, Shandong University, Jinan, Shandong 250100, China
2School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Received 20 February 2013; Accepted 20 April 2013
Academic Editor: Guangchen Wang
Copyright © 2013 Shaolin Ji and Shuzhen Yang. 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.
In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.
It is well known that quasilinear parabolic partial differential equations are related to Markovian forward-backward stochastic differential equations (see [1–3]), which generalizes the classical Feynman-Kac formula. Recently in the framework of functional Itô calculus, a path-dependent PDE was introduced by Dupire  and the so-called functional Feynman-Kac formula was also obtained. For a recent account and development of this theory we refer the reader to [5–11].
In this paper, we study a functional forward-backward system and its relation to a quasilinear parabolic path-dependent PDE. In more details, the functional forward-backward system is described by the following forward-backward SDE:
Equation (1) is an uncoupled functional forward-backward system, its general the results of , and there are many applications of the uncoupled functional forward-backward system in optimal control problem. The main difference is that we give a weaker requirement of and about , and we also establish some estimates and regularity results for the solution with respect to paths. Then, we prove that the solution of (1) is the unique classical solution of the following path-dependent PDE: where
The paper is organized as follows: in Section 2, we give the notations and results on functional FBSDEs and functional Itô calculus. Some estimates and regularity results for the solution of FBSDEs are established in Section 3. Finally, we prove the relationship between functional FBSDEs and path-dependent PDEs in Section 4.
2.1. Functional FBSDEs
Let and let be the Wiener measure on . We denote by the cannonical Wiener process, with , , . For any we denote by the -completion of .
For any , we denote by the set of all square integrable -measurable random variables, the set of all -valued -adapted processes such that
Let and . For every , we consider the following functional forward-backward SDEs: where The processes take values in , , ; , , , and take values in , , , and . Equations (5) and (6) can be rewritten as For , we define . For , , and for ,
We give the following assumption.
Assumption 1. For all , , , and , there exists a constant , such that and for all ,
Then we have the following theorem (see ).
2.2. Functional Itô Calculus
The following notations and tools are mainly from Dupire . Let be fixed. For each , we denote by the set of càdlàg -valued functions on . For each the value of at time is denoted by . Thus is a càdlàg process on and its value at time is . The path of up to time is denoted by , that is, . We denote . For each and we denote by the value of at and , which is also an element in .
Let and denote the inner product and norm in . We now define a distance on . For each and , we denote It is obvious that is a Banach space with respect to and is not a norm.
Definition 4. A function is said to be -continuous at , if for any there exists such that for each with , we have . is said to be -continuous if it is -continuous at each .
Definition 5. Let and be given. If there exists , such that
then we say that is vertically differentiable at and denote the gradient of . If exists for each , is said to be vertically differentiable in .
We can similarly define the Hessian . It is an -valued function defined on , where is the space of all symmetric matrices.
For each we denote It is clear that .
Definition 6. For a given if we have then we say that is (horizontally) differentiable in at and . is said to be horizontally differentiable in if exists for each .
Definition 7. Define as the set of function defined on which are times horizontally and times vertically differentiable in such that all these derivatives are -continuous.
Theorem 8 (functional Itô’s formula). Let be a probability space, if is a continuous semi-martingale and is in , then for any ,
We first recall some notions in Pardoux and Peng . , , will denote, respectively, the set of functions of class from into , the set of those functions of class whose partial derivatives of order less than or equal to are bounded, and the set of those functions of class which, together with all their partial derivatives of order less than or equal to , grow at most like a polynomial function of the variable at infinity.
Now we give the definition of derivatives in our context. Under Assumption 1 we have that has a unique solution. For , set Then the following definition of derivatives will be used frequently in the sequel.
Definition 9. An -valued function is said to be in , if for and , there exist and ( is the set of all order symmetric matrix) such that We denote , and . is said to be in if and exist for each , and there exist some constants and depending only on such that for each , and for each , with . We can also define , , , and , , , .
Now we consider the solvability of (6).
Assumption 10. Let be an -valued function on . Moreover with the Lipschitz constants and .
Assumption 11. Let , where is such that is of class and the first order partial derivatives in , , and are bounded, as well as their derivatives of up to order two with respect to , .
It is obvious under Assumptions 1, 10, and 11 the FBSDE (5) and (6) has a unique solution (see [12–14]).
3.1. Regularity of the Solution of FBSDEs
Proof. To simplify presentation, we only study the case , and .
The application of Itô's formula to yields that So Then we have Applying Itô's formula to yields that By inequality and Burkholder-Davis-Gundy's inequality, there is a such that By Assumption 1 and Gronwall's inequality, from (29) we have (note that will change line by line) By Assumptions 10 and 11 and taking , from (27) we have where . Similarly we can get the same result for .
This completes the proof.
Proof. can be formed as a linearized BSDE: for each ,
where (with )
Under Assumptions 10, 11, using the same method as in Lemma 12, we get the first three inequalities.
For the next three inequalities, we write as the solution of the following linearized BSDE: Then the same calculus implies that Consider Set Then it solves the following BSDE: where Thus, under Assumptions 10, 11, similarly as in Lemma 12, we can get the last three inequalities.
Theorem 14. For each , has a version which is a.e. of class .
Proof. We only consider one dimensional case. Applying Lemma 12, for each and , By Kolmogorov's criterion, there exists a continuous derivative of with respect to . There also exists a mean-square derivative of with respect to , which is mean square continuous in . We denote them by By Theorem 13 and Definition 9, is the solution of the following BSDE: It is easy to check that the above BSDE has a unique solution. Thus the existence of a continuous second order derivative of with respect to is proved in a similar way.
Define We have the following results about .
Lemma 15. For all , one has .
Proof. For given , , set . Consider the solutions of FBSDE (5) and (6) on :
We need to prove .
For each fixed and positive integer , we introduce a mapping where , ,
Denote Set where is a division of , , . For any , is the solution of the following BSDE: Multiplying by and adding the corresponding terms, we obtain By the uniqueness and existence theorem of BSDE, we have Then, by the definition of , we get Note that This completes the proof.
Corollary 16. is -continuous and exist; moreover they are both -continuous.
Proof. By Theorem 14 we know that and exist. In the following, we only prove is -continuous. The proof for the continuous property of and is similar. Taking expectation on both sides of (6), For , we have By Theorem 13, for some constant depending on , and , This completes the proof.
3.2. Path Regularity of Process
In Pardoux and Peng , BSDE is only state-dependent, that is, and . Under appropriate assumptions, and are related in the following sense: Under the conditions and in , Peng and Wang extend this result to the path-dependent case. The corresponding BSDE is where Then under some assumptions, they obtained In our context, we have the following theorem.
To prove the above theorem, we need the following lemma essentially from Pardoux and Peng .
Lemma 18. Let and some be given. Suppose that where is in . For , suppose that where . Then for each ,
Proof. We only consider the one dimensional case.
For , the FBSDE (5), (6) can be rewritten as For ,
Now consider the following system of quasilinear parabolic differential equations, which is defined on and parameterized by : where . The other one is defined on : where . By Corollary 16 and Theorems 3.1, 3.2 in Paroux-Peng , we have , , and Then we obtain Finally, for each , In particular, This completes the proof.
Now we give the proof of Theorem 17.
Proof. For each fixed and positive integer , we introduce a mapping where , For each , there exist some functions defined on and defined on such that Indeed, if we set