Pure and Applied Mathematics Journal
Volume 4, Issue 3, June 2015, Pages: 101-108

Maximum Principle and the Applications of Mean-Field Backward Doubly Stochastic System

Hong Zhang1, Jingyi Wang2, Tengyu Zhao3, Li Zhou1

1School of Information, Beijing Wuzi University, Beijing, China

2School of Banking and Finance, University of International Business and Economics, Beijing, China

3School of Management Science and Engineering, Central University of Finance and Economics, Beijing, China

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(Hong Zhang)

To cite this article:

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 , . 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 established9-10

(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)


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