The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations
Jie Zhu^{1}, Hong Zhang^{1}, Li Zhou^{1}, Yuhang Feng^{2}
^{1}School of Information, Beijing Wuzi University, Beijing, China
^{2}Insurance Department, Central University of Finance and Economics, Beijing, China
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To cite this article:
Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng. The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 120-127. doi: 10.11648/j.pamj.20150403.20
Abstract: Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.
Keywords: FBSDE, Mean-Field Forward Backward, Stochastic Differential Equations, Stochastic Partial Differential Equations
1. Introduction
In order to study the stochastic partial differential equations are non local (SPDEs):
(1)
Where
In this section we study the mean field forward backward stochastic differential equations (MF-FBDSDE)
and
(2)
Equation (2) to Carmona and Delarue's results are extended to the stochastic case, the Peng and Shi results are extended to the case of the average field. Under certain monotonicity conditions, through the continuity of MF-FBDSDE (2) method to get the existence and uniqueness of the solution. Finally, the application of WF-FBDSDE (2), are non stochastic partial differential equation (1) represents the local solution of the probability.
2. Problem Presentation
Hypothesis is a complete space with its own product space here, for any completion. Arbitrary definition of in the can be extended to natural space, namely . , and so on, we define
When at the same time,
In order to distinguish between and, we use mark E' and E*.
From now on, when we talk about MF-BDSVIE , the mapping of and are defined by the E' operator. Obviously, is a non local means the value in the depends on the whole set
Not only
Introduction of mark
Definition. is called the solution of MF-FBDSDEs if is meeting (2)
We assume that for any (H1) , it is meeting
(H2) is meeting Lipschitz condition: there exist contact , making
(H3) are meeting the following monotonicity condition, that is to say, for constant, making
Where
3. The Existence and Uniqueness of Solutions for MF-FBDSDEs
In order to hypothesis (H1) -(H3), prove the equation (3.68) the existence and uniqueness of the solution, we need the following lemma. The following lemma is discussed in class. is a priori parameter estimation of MF-FBDSDEs:
(3)
Where , , are arbitrary given vector-valued random variables.
When = 1, the existence of solution of equation (3.69) means that the existence of solution of equation (3.68) exist. When = 0, according to the [111] of the MF - BDSDE results about the existence and uniqueness of the solution, that the equation (3.69) is the only solution can be got.
The following lemma is the key in the continuous method, it provides for a fixed , if the equation (3.69) is the only solution, there exist a has nothing to do with the normal number , to make the equation (3.69) is the only solution also for .
Lemma 2. Supposing that (H1)-(H3), if the equation (3.69) is the only solution for certain ,,,
There exist a has nothing to do with the normal number, to make the equation (3.69) is the only solution also in the for , , .
Proof
Supposing that
When , if the equation (3.69) is the only solution for ,,, there exist only meeting the following equations for each .
Where is independent of , Our aim is to prove the following mapping
It can be compressed for small enough
Supposing that
On the , The formula was applied with ,we can obtain
Where
according to the H1-H3,
Where constant C>0, thereafter, it would be appropriate constants. It can be progressive different and only depends on the Lipschitz constant. On the other hand, we apply estimation technology to . We apply formula to on the .we can get the following,
Where
Form the (H.3), we can get the following
Thus
Form Gronwall inequality, we can obtain the following
Combined with the above estimates (3.70) and (3.71), The constant of the full C>0, the following is available.
It is easy to see, for each fixed , the mapping is compressed, that is to say,
From that, we could know
Thus the existence of the fixed point of mapping, conclusion is proved.
Existence and uniqueness theorem of solutions of MF-FBDSDE is given below.
Theorem 3. We suppose the (H1)-(H3) are tenable, thus MF-FBDSDE exist uniqueness solutions in the
.
Proof
(Uniqueness) We take and as too solutions of the equation (3.68). We continue to use the mark in 3.5.2 lemma. Application formula. to on
We can get,
According to assumption (H.3), we can know
So , Uniqueness is proved.
(Existence) when ,The equation(3.69)has only one solution in
According to Lemma3.5.2, we can know
There exists positiveso to any
When , The equation(3.69)has only one solution. Because only relies on , Repeat the above process many times, Leading to.Particularly, when ,take, , The equation(3.69)has only one solution in
Conclusion is proved.
Note: hypothesis (H.3) could be replaced by the following
Where and both are positive constant.
Theorem 5. We suppose the (H1), (H2) and (H3)’ are tenable, thus MF-FBDSDE exist uniqueness solutions in the
.
4. Probabilistic Representations of Non Local SPDEs Solutions
Using the above MF-FBDSDEs, discuss the non probabilistic local SPDE solutions. For any , consider the following WF-FBDSDE:
We suppose that of MF-FBDSDE is deterministic, there exist uniqueness solution of MF-FBDSDE. We suppose that
According the uniqueness solution of MF-FBDSDE, we know that for any , we could obtain the following,
In order to mark is simple, for we suppose that
Notice the marks in the second quarter, we know that
If there exist A is the following second-order quasilinear nonlocal SPDE:
We can obtain the following:
Theorem3.5.6.Suppose is deterministic in MF-FBDSDE(3.72).
There is only one solution in MF-FBDSDE(3.72).F,f,G and g are three order continuous differentiable.is two order continuous differentiable.
If is the solution of nonlocal SPDE(3.74),so(3.73)is right.(Y,Z)is only determined by equation(3.72)
Proof We only need to prove is the solution of MF-BDSDE. Thus we suppose, we can obtain the following
Here we used to meet the Ito formula and U condition equation. Finally, the cell length tends to 0. We can obtain the following
It is easy to verify have the same solution with MF-BDSDE.
Note 3.5.7. (I) in non local SPDE, When p=0, the non local SPDE (3.73) degradation as follows
II) (3.73) can be called non local SPDE (3.74) Feynman-Kac formula.
III) (3.73) to PDE with algebraic equations to the mean field.
Acknowledgements
This paper is funded by the project of National Natural Science Fund, Logistics distribution of artificial order picking random process model analysis and research (Project number: 71371033); and funded by intelligent logistics system Beijing Key Laboratory (No.BZ0211); and funded by scientific-research bases--- Science & Technology Innovation Platform---Modern logistics information and control technology research (Project number: PXM2015_014214_000001); and funded by 2014-2015 school year, Beijing Wuzi University, College students' scientific research and entrepreneurial action plan project (No.68); and funded by Beijing Wuzi University, Yunhe scholars program (00610303/007); and funded by Beijing Wuzi University, Management science and engineering Professional group of construction projects. (No. PXM2015_014214_000039). University Cultivation Fund Project of 2014-Research on Congestion Model and algorithm of picking system in distribution center (0541502703)
References