Maximum Principle and the Applications of Mean-Field Backward Doubly Stochastic System
Hong Zhang^{1}, Jingyi Wang^{2}, Tengyu Zhao^{3}, Li Zhou^{1}
^{1}School of Information, Beijing Wuzi University, Beijing, China
^{2}School of Banking and Finance, University of International Business and Economics, Beijing, China
^{3}School of Management Science and Engineering, Central University of Finance and Economics, Beijing, China
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Hong Zhang, Jingyi Wang, Tengyu Zhao, Li Zhou. Maximum Principle and the Applications of Mean-Field Backward Doubly Stochastic System. Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 101-108. doi: 10.11648/j.pamj.20150403.17
Abstract: Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.
Keywords: Mean-Field Backward Doubly, Stochastic System, Stochastic Control
1. Introduction
Andersson and Djehiche, Buckdahn, Djehiche and Li, Meyer Brandis, ksendal and Zhou, and Lihave studied the optimal control problem about Mean-field s tochastic differential system .Inspired by the above problems, in the paper, we study the optimal control problem about Mean-field backward doubly stochastic system. In the situation that control field to the convex and coefficient contains control variable, Using convex variational and dual technology, we present the local and global stochastic maximum principle, proved a sufficient conditions of optimality (verification theorem) and a necessary condition[1-4].
2. The Control Problem of Mean-Field Backward Doubly Stochastic System
For simple marking, make . Given convex subset , allowing the control set is defined as
is - measurable,
For any consider the following MF - BDSDE:
Where
And ,
Performance indicators is
(1)
Where
Control problem is looking for admission control to make performance indicators reaching the minimum value on the.
Supposing that [5-6]
(H1) (1) is continuously differentiable about ,and the derivative of h and i is linear growth.
(2) meet uniform Lipschitz condition about .
In other words there exist for . making
And
Where .
Under the above assumptions, for any, there exists a unique solution of the equation (1). And the performance index defined is reasonable.[7-8]
Assumed is the optimal control. is the corresponding optimal trajectory. meet .because of the convexity of , for any . there exists a unique solution of
Lemma 1. hypothesis (H1) is established, for any [0, T],
Proof. Notice that to meet the following MF-BDSDE:
Applying formulas to
According to (H1), there is
Where is constant rely on (H1). According Gronwall Inequality and Burkholder-Davis-Gundy Inequality, results are verified.
For simple marking, make
(2)
Where
And
Marked
Under the above assumptions, for any, there exists a unique solution of the equation (2).
Lemma 2. Marked
,
(3)
is the solution of the equation as follows,
Where , ,
and
Applying formulas to on
According to (H1), there is
Where is constant, when 时, . According Gronwall Inequality, results are verified. Because of is the optimal control,
(4)
According to lemma 2, there is
lemma 3. Hypothesis (H1) was established, then the following variation inequality is established^{9-10}：
(5)
Where
Proof.
Where .
so (5) is verified.
Considering the adjoin equation:
(6)
Where
Define the Hamiltonian function as follows
(7)
By the variational inequality (7), we present MF - BDSDEs stochastic control problem of stochastic maximum principle.
Theorem 1,(stochastic maximum principle) Assumed is the optimal trajectory of the control problem{(1),(2)},
(8)
where
(9)
Proof. Applying formulas to,we can get
According (5) ,we can get
According Hamiltonian function, we can get
For any is the any element of ,setting
We can know , because meet , setting , The above inequalities can be rewritten as
Differential on a variable at , we can get
So (8) is verified.
3. Mean-Field Backward Doubly Stochastic LQ Problem
This section, we apply the maximum value principle to Mean-field backward doubly stochastic LQ problem.
And
Where is bounded. is measurable (similarly, other coefficient satisfies the hypothesis). are nonnegative, is positive. The state equation is
(10)
Performance indicators is
In order to mark is simple, put for . Hamiltonian function is
According Theorem 1, we can get
(11)
Where
And
(12)
And
Theorem 2. Assumes that satisfy (9), the (p, g) is the solution of equation (10), the above LQ problem have unique solution
Proof.
Applying formulas to on
We can get
Theorem 2 is verified.
4. Summary
Theorem 1,(stochastic maximum principle) Assumed is the optimal trajectory of the control problem{(1),(2)},
(13)
where
(14)
Theorem 2. Assumes that satisfy (9), the (p, g) is the solution of equation (10), the above LQ problem have unique solution.
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