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

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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Analysis and control for transient responses of seismic-excited hysteretic structures

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SOIL dynamicS AND EARTHQUAKE ENGINEERING 2015年 73卷 58-65页

作者： Jin, X. L. Wang, Y. Huang, Z. L. Zhejiang Univ Dept Engn Mech State Key Lab Fluid Power Transmiss & Control Hangzhou 310027 Zhejiang Peoples R China

The semi-arialytical solution of transient responses and the bounded control strategy to minimize the transient responses for seismic-excited hysteretic structures are investigated in this manuscript, the hysteretic behavior is described by Duhem model while the seismic excitations by random processes with Kanai-Tajimi spectrum. The averaged Fokker-Planck-Kolmogorov equation with respect to the probability density of amplitude response is firstly derived by utilizing the stochastic averaging technique based on the generalized harmonic functions. The probability density is approximately expressed as a series expansion in terms of a set of specified basis functions with time-dependent coefficients which are determined through the Galerkin procedure. The quasi-optimal bounded control strategy to minimize the transient response is proposed based on the averaged system with respect to amplitude response and an appropriate performance index. The quasi-optimal control is derived from the minimum condition in the dynamic programming equation. The application and effectiveness of the proposed analytical procedure and control strategy are illustrated through one representative example. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Hysteretic structure Transient response Bounded control Seismic excitation Stochastic averaging Galerkin procedure dynamic programming principle

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Viscosity Solutions of Hybrid Game Problems with Unbounded Cost Functionals

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INTERNATIONAL GAME THEORY REVIEW 2016年第1期18卷 1550016-1550016页

作者： Sheetal, Dharmatti Indian Inst Sci Educ & Res Thiruvananthapuram Sch Math Comp Sci & Engn Dept CET Campus Thiruvananthapuram 695016 Kerala India

This paper analyzes zero sum game involving hybrid controls using viscosity solution theory where both players use discrete as well as continuous controls. We study two problems, one in finite horizon and other in infinite horizon. In both cases, we allow the cost functionals to be unbounded with certain growth, hence the corresponding lower and upper value functions defined in Elliot-Kalton sense can be unbounded. We characterize the value functions as the unique viscosity solution of the associated lower and upper quasi variational inequalities in a suitable function class. Further we find a condition under which the game has a value for both games. The major difficulties arise due to unboundedness of value function. In infinite horizon case we prove uniqueness of viscosity solution by converting the unbounded value function into bounded ones by suitable transformation. In finite horizon case an argument is based on comparison with a supersolution.

关键词： dynamic programming principle viscosity solutions game theory hybrid games

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Dividend optimization under reserve constraints for the Cramer-Lundberg model compounded by force of interest

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ECONOMIC MODELLING 2015年 46卷 142-156页

作者： Zhu, Jinxia Chen, Feng Univ New S Wales Sch Risk & Actuarial Studies Sydney NSW 2052 Australia Univ New S Wales Sch Math & Stat Sydney NSW 2052 Australia

We study the dividend optimization problem for a company where surplus in the absence of dividend payments follows a Cramer-Lundberg process compounded by constant force of interest. The company controls the times and amounts of dividend payments subject to reserve constraints that dividends are not payable if the surplus is below b(0) and that a dividend payment, if any, cannot reduce the surplus to a level below b(0), and its objective is to maximize the expected total discounted dividends. We show how the optimality can be achieved under the constraints and construct an optimal strategy of a band type. (c) 2014 Elsevier B.V. All rights reserved.

关键词： Cramer-Lundberg model dynamic programming principle Hamilton-Jacobi-Bellman equation Optimal dividend strategy Solvency constraints

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Using Approximate dynamic programming for Estimating the Revenues of a Hydrogen-based High-Capacity Storage Device

Using Approximate Dynamic Programming for Estimating the Rev...

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IEEE Symposium on Adaptive dynamic programming and Reinforcement Learning (ADPRL)

作者： Francois-Lavet, Vincent Fonteneau, Raphael Ernst, Damien Univ Liege Dept Elect Engn & Comp Sci B-4000 Liege Belgium

ISBN: (纸本)9781479945528

This paper proposes a methodology to estimate the maximum revenue that can be generated by a company that operates a high-capacity storage device to buy or sell electricity on the day-ahead electricity market. The methodology exploits the dynamic programming (DP) principle and is specified for hydrogen-based storage devices that use electrolysis to produce hydrogen and fuel cells to generate electricity from hydrogen. Experimental results are generated using historical data of energy prices on the Belgian market. They show how the storage capacity and other parameters of the storage device influence the optimal revenue. The main conclusion drawn from the experiments is that it may be advisable to invest in large storage tanks to exploit the inter-seasonal price fluctuations of electricity.

关键词： dynamic programming electrolysis fuel cells hydrogen storage power markets Belgian market day-ahead electricity market dynamic programming principle high-capacity storage device hydrogen-based storage devices interseasonal price fluctuations maximum revenue estimation optimal revenue dynamic programming Electricity Electrochemical processes Fuel cells Hydrogen Hydrogen storage

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Discontinuous control problems with state constraints: Linear formulations and dynamic programming principles

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2013年第2期402卷 635-647页

作者： Goreac, Dan Ivascu, Carina Univ Paris Est LAMA CNRS UMR 8050UPEMLVUPEC F-77454 Marne La Vallee France Univ Transilvania Fac Matemat Informat Brasov Romania

This paper aims at studying a class of discontinuous deterministic control problems under state constraints using a linear programming approach. As for classical control problems (Gaitsgory and Quincampoix (2009) [16], Goreac and Serea (2011) [19]), the primal linear problem is stated on some appropriate space of probability measures. Naturally, the support of these measures is contained in the set of constraints. This linearized value function and its dual can, alternatively, be seen as the limit of standard penalized problems. Second, we provide a semigroup property for this set of probability measures leading to dynamic programming principles for control problems under state constraints. An abstract principle is provided for general bounded cost. Linearized versions are obtained under further (semi)continuity assumptions. (c) 2012 Elsevier Inc. All rights reserved.

关键词： Control under state constraints dynamic programming principle Linear programming Hamilton-Jacobi equations

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Min-max control problems via occupational measures

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OPTIMAL CONTROL APPLICATIONS & METHODS 2014年第3期35卷 340-360页

作者： Goreac, Dan Serea, Oana-Silvia Univ Paris Est CNRS UPEC LAMA UMR 8050UPEMLV F-77454 Marne La Vallee France Univ Perpignan F-66000 Perpignan France

We propose a linearized formulation for min-max control problems with separated dynamics. First, we investigate the existence of the value function and saddle points for semicontinuous costs. Second, we obtain dual fo... 详细信息

关键词： dynamic programming principle min-max control problems linearization techniques occupational measures

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OPTIMAL CONTROL PROBLEMS OF FULLY COUPLED FBSDEs AND VISCOSITY SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2014年第3期52卷 1622-1662页

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

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton-Jacobi-Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85-138. We emphasize that when sigma depends on the second component of the solution (Y, Z) of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence and uniqueness result for fully coupled FBSDEs when the Lipschitz constant of sigma with respect to z is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.

关键词： fully coupled FBSDEs value functions stochastic backward semigroup dynamic programming principle viscosity solution

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SOBOLEV WEAK SOLUTIONS OF THE HAMILTON-JACOBI-BELLMAN EQUATIONS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2014年第3期52卷 1499-1526页

作者： Wei, Lifeng Wu, Zhen Zhao, Huaizhong Ocean Univ China Sch Math Sci Qingdao 266003 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China Univ Loughborough Dept Math Sci Loughborough LE11 3TU Leics England

This paper is concerned with the Sobolev weak solutions of the Hamilton-Jacobi-Bellman (HJB) equations. These equations are derived from the dynamic programming principle in the study of stochastic optimal control problems. Adopting the Doob-Meyer decomposition theorem as one of the main tools, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. In the recursive optimal control problem, the cost function is described by the solution of a backward stochastic differential equation (BSDE). This problem has a practical background in economics and finance. We prove that the value function is the unique Sobolev weak solution of the related HJB equation by virtue of the nonlinear Doob-Meyer decomposition theorem introduced in the study of BSDEs.

关键词： dynamic programming principle Hamilton-Jacobi-Bellman equations Sobolev weak solutions backward stochastic differential equations Doob-Meyer decomposition theorem

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A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

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APPLIED MATHEMATICS AND OPTIMIZATION 2014年第2期70卷 253-278页

作者： Hu, Mingshang Ji, Shaolin Yang, Shuzhen Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China Shandong Univ Qilu Inst Finance Jinan 250100 Shandong Peoples R China

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

关键词： G-expectation Backward stochastic differential equations Stochastic optimal control dynamic programming principle Viscosity solution

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