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FULLY COUPLED FORWARD-BACKWARD SDES INVOLVING THE VALUE FUNCTION AND ASSOCIATED NONLOCAL HAMILTON-JACOBI-BELLMAN EQUATIONS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2016年第2期22卷 519-538页

作者： Hao, Tao Li, Juan Shandong Univ Sch Math & Stat Weihai 264209 Peoples R China Shandong Univ Tradit Chinese Med Sch Sci & Technol Jinan 250355 Peoples R China

A new type of controlled fully coupled forward-backward stochastic differential equations is discussed, namely those involving the value function. With a new iteration method, we prove an existence and uniqueness theorem of a solution for this type of equations. Using the notion of extended "backward semigroup", we prove that the value function satisfies the dynamic programming principle and is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman equation.

关键词： Fully coupled FBSDE involving value function dynamic programming principle fully coupled mean-field FBSDE viscosity solution nonlocal Hamilton-Jacobi-Bellman equation

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BSDES IN GAMES,COUPLED WITH THE VALUE *** NONLOCAL BELLMAN-ISAACS EQUATIONS

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Acta Mathematica Scientia 2017年第5期37卷 1497-1518页

作者：郝涛李娟 School of Statistics Shandong University of Finance and Economics School of Mathematics and Statistics Shandong University

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

关键词： Mc Kean-Vlasov SDE BSDE coupled with the lower and the upper value functions dynamic programming principle mean-field BSDE viscosity solution coupled nonlocal HJB-Isaacs equation Isaacs' condition

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Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2008年第1期47卷 444-475页

作者： Buckdahn, Rainer Li, Juan Univ Bretagne Occidentale Dept Math F-29285 Brest France Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Shandong Univ Weihai Dept Math Weihai 264200 Peoples R China

In this paper we study zero-sum two-player stochastic differential games with the help of the theory of backward stochastic differential equations (BSDEs). More precisely, we generalize the results of the pioneering work of Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game. This extension of the class of admissible control processes has the consequence that the cost functionals become random variables. However, by making use of a Girsanov transformation argument, which is new in this context, we prove that the upper and the lower value functions of the game remain deterministic. Apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information, its combination with BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng [BSDE and stochastic optimizations, in Topics in Stochastic Analysis, Science Press, Beijing, 1997], allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way. The upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is extended from the framework of stochastic control theory into that of stochastic differential games.

关键词： stochastic differential games value function backward stochastic differential equations dynamic programming principle viscosity solution

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Stochastic representation for solutions of Isaacs' type integral-partial differential equations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2011年第12期121卷 2715-2750页

作者： Buckdahn, Rainer Hu, Ying Li, Juan Shandong Univ Weihai Sch Math & Stat Weihai 264200 Peoples R China Univ Bretagne Occidentale Dept Math F-29238 Brest 3 France Univ Rennes 1 IRMAR F-35042 Rennes France Shandong Univ Inst Adv Study Jinan 250100 Peoples R China

In this paper we study the integral partial differential equations of Isaacs' type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral partial differential equations of Hamilton Jacobi Bellman Isaacs' type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs' condition. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Stochastic differential games Poisson random measure Value function Backward stochastic differential equations dynamic programming principle Integral partial differential operators Viscosity solution

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Stochastic Maximum principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

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Chinese Annals of Mathematics,Series B 2018年第5期39卷 773-790页

作者： Siyu LV Zhen WU School of Mathematics Southeast UniversityNanjing 211189China School of Mathematics Shandong UniversityJinan 250100China

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.

关键词： Stochastic maximum principle dynamic programming principle Forward-backward stochastic differential equation Regime switching Jump diffusion

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Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

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APPLIED MATHEMATICS AND OPTIMIZATION 2015年第3期71卷 411-448页

作者： Li Juan Wei Qingmeng Shandong Univ Sch Math & Stat Weihai 264209 Weihai Peoples R China NE Normal Univ Sch Math & Stat Changchun 130024 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715-2750, 2011), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs' type combined with an algebraic equation, respectively. Furthermore, for a special case (when and do not depend on ), under the Isaacs' condition, we get the existence of the value of the game.

关键词： Fully coupled FBSDEs with jumps Stochastic differential game Hamilton-Jacobi-Bellman-Isaacs equation Value function Stochastic backward semigroup dynamic programming principle Viscosity solution

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Saddle points of discrete Markov zero-sum game with stopping

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AUTOMATICA 2012年第8期48卷 1898-1903页

作者： Li, Xun Shen, Jie Song, Qingshuo Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

We study the sufficient conditions for the existence of a saddle point of a time-dependent discrete Markov zero-sum game up to a given stopping time. The stopping time is allowed to take either a finite or an infinite non-negative random variable with its associated objective function being well-defined. The result enables us to show the existence of the saddle points of discrete games constructed by Markov chain approximation of a class of stochastic differential games. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Saddle points dynamic game Markov chain approximation dynamic programming principle

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Path-dependent Hamilton-Jacobi-Bellman equations related to controlled stochastic functional differential systems

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OPTIMAL CONTROL APPLICATIONS & METHODS 2015年第1期36卷 109-120页

作者： Ji, Shaolin Wang, Lin Yang, Shuzhen Shandong Univ Inst Financial Studies Jinan 250100 Shandong Peoples R China Shandong Univ Inst Math Jinan 250100 Shandong Peoples R China Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

In this paper, a stochastic optimal control problem is investigated in which the system is governed by a stochastic functional differential equation. In the framework of functional Ito calculus, we build the dynamic programming principle and the related path-dependent Hamilton-Jacobi-Bellman equation. We prove that the value function is the viscosity solution of the path-dependent Hamilton-Jacobi-Bellman equation. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： stochastic functional differential equations dynamic programming principle viscosity solution path-dependent HJB equations

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Singular linear quadratic optimal control for singular stochastic discrete-time systems

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OPTIMAL CONTROL APPLICATIONS & METHODS 2013年第5期34卷 505-516页

作者： Feng, Jun-e Cui, Peng Hou, Zhongsheng Shandong Univ Sch Math Jinan 250100 Peoples R China Shandong Univ Sch Control Sci & Engn Jinan 250061 Peoples R China Beijing Jiaotong Univ Adv Control Syst Lab Beijing 100044 Peoples R China

The finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete-time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright (c) 2012 John Wiley & Sons, Ltd.

关键词： dynamic programming principle LQ optimal control singular systems stochastic systems

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A GAME INTERPRETATION OF THE NEUMANN PROBLEM FOR FULLY NONLINEAR PARABOLIC AND ELLIPTIC EQUATIONS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2013年第4期19卷 1109-1165页

作者： Daniel, Jean-Paul Univ Paris 06 LJLL CNRS UMR 7598 F-75005 Paris France

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter e which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as e tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.

关键词： Fully nonlinear elliptic equations viscosity solutions Neumann problem deterministic control optimal control dynamic programming principle oblique problem mixed-type Dirichlet-Neumann boundary conditions

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