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THE EXISTENCE AND UNIQUENESS OF VISCOSITY SOLUTION TO A KIND OF HAMILTON-JACOBI-BELLMAN EQUATION

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2019年第6期57卷 3911-3938页

作者： Hu, Mingshang Ji, Shaolin Xue, Xiaole Shandong Univ Zhongtai Secur Inst Financial Studies Jinan 250100 Shandong Peoples R China Shandong Univ Sch Management Jinan 250100 Peoples R China

In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equation combined with algebra equations. This HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn, and Pardoux [Stochastics, 60 (1997), pp. 57-83] usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this HJB equation. We obtain that the value function is the minimum viscosity solution to this HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.

关键词： dynamic programming principle fully coupled forward-backward stochastic differential equations Hamilton-Jacobi-Bellman equation viscosity solution

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On Tauberian theorem for stationary Nash equilibria

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OPTIMIZATION LETTERS 2019年第8期13卷 1855-1870页

作者： Khlopin, D. V. Russian Acad Sci Krasovskii Inst Math & Mech 16 S Kovalevskaja St Ekaterinburg 620990 Russia Ural Fed Univ Inst Math & Comp Sci Chair Appl Math & Mech 4 Turgeneva St Ekaterinburg 620083 Russia

We consider general n-player nonzero-sum dynamic games, which is broader than differential games and could accommodate both discrete and continuous time. Assuming common dynamics, we study the long run average family and discounting average family of the running costs. For each of these game families, we investigate asymptotic properties of its Nash equilibria. We analyze asymptotic Nash equilibria-strategy profiles that are approximately optimal if the planning horizon tends to infinity in long run average games and if the discount tends to zero in discounting games. Moreover, we also assume that this strategy profile is stationary. Under a mild assumption on players' strategy sets, we prove a uniform Tauberian theorem for stationary asymptotic Nash equilibrium. If a stationary strategy profile is an asymptotic Nash equilibrium and the corresponding Nash value functions converge uniformly for one of the families (when discount goes to zero for discounting games, when planning horizon goes to infinity in long run average games), then for the other family this strategy profile is also an asymptotic Nash equilibrium, and its Nash value functions converge uniformly to the same limit. As an example of application of this theorem, we consider Sorger' model of competition of two firms.

关键词： Stationary Nash equilibrium Tauberian theorem Abel mean Cesaro mean dynamic programming principle Uniform value

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Research on investment incorporating both environmental performance and long (short) term financial performance of firms

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2024年第2期55卷 273-299页

作者： Li, Liang Fei, Chen Fei, Weiyin Anhui Polytech Univ Sch Math Phys & Finance Wuhu Peoples R China Univ Shanghai Sci & Technol Business Sch Shanghai Peoples R China Univ Shanghai Sci & Technol Business Sch Shanghai 200093 Peoples R China

Firstly, with the rapid popularisation of the ESG concept and the deepening of the conception of green and low-carbon development, the performance of a firm in protecting and improving the ecological environment and the level of green technology investment has gradually become the reference criteria for judging whether the firm has social value and sustainability in development. Secondly, to pursue short-term survival and long-term development, the firm will also vigorously strive for short-term profit creation and long-term capital accumulation. Thirdly, due to the existence of information asymmetry, the principal-agent parties in a firm may generate agency conflicts in the pursuit of maximising their own interests, which may lead to losses or even bankruptcy. Therefore, making rational short-term, long-term, and green technology investment decisions that consider both characteristics of firm and agency conflicts is essential for the survival and sustainable development of the firm. From the perspective of contract theory, this paper considers the design of an optimal dynamic financial contract that considers the environment's improvement and the achievement of the firm's long- and short-term financial performance while satisfying incentive compatibility. On this basis, we explore how the optimal investment strategies vary with firm characteristics. The results of the study suggest that the decisions of green technology investment, short-term investment, and long-term capital investment should be appropriately adjusted according to the level of financial slack and the effect of different market shocks to facilitate the achievement of the dual objectives of long- and short-term profitability and environmental improvement.

关键词： Green technology investment (GTI) environmental performance (EP) sustainable development dynamic financial contract dynamic programming principle

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A PSEUDO-MARKOV PROPERTY FOR CONTROLLED DIFFUSION PROCESSES

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2016年第2期54卷 1017-1029页

作者： Claisse, Julien Talay, Denis Tan, Xiaolu INRIA Sophia Antipolis 2004 Route LuciolesBP93 Sophia Antipolis France Univ Paris 09 CEREMADE Pl Marechal De Lattre De Tassigny F-75775 Paris 16 France

In this note, we propose two different approaches to rigorously justify a pseudo-Markov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314]. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.

关键词： stochastic control martingale problem dynamic programming principle pseudo-Markov property

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Ergodic control of reflected diffusions with jumps

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APPLIED MATHEMATICS AND OPTIMIZATION 1997年第2期35卷 117-137页

作者： Menaldi, JL Robin, M CERN EUROPEAN LAB PARTICLE PHYSCH-1211 GENEVA 23SWITZERLAND

We discuss ergodicity properties of a controlled jumps diffusion process reflected from the boundary of a bounded domain. The control parameters act on the drift term and on a first-order-type jump density. The controlled process is generated via a Girsanov change of probability, and a long-run average criterion is optimized. An optimal stationary feedback is constructed by means of the Hamilton-Jacobi-Bellman equation.

关键词： reflected jump diffusion ergodic optimal control dynamic programming principle Hamilton-Jacobi-Bellman equation Green function Girsanov transformation Doeblin condition discounted control problem

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Mean-field backward stochastic differential equations and related partial differential equations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2009年第10期119卷 3133-3154页

作者： Buckdahn, Rainer Li, Juan Peng, Shige Shandong Univ Sch Math & Stat Weihai 264200 Peoples R China Univ Bretagne Occidentale Dept Math F-29285 Brest France Shandong Univ Sch Math Jinan 250100 Peoples R China

In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Arm. Probab. (2007) (in press). Available online: http://***/aop/future_papers. h(m] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of "particles" (or "agents"). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we Study a mean-field BSDE in a Markovian framework, associated with a McKean-Vlasov forward equation. By combining classical BSDE methods, in particular that of "backward semigroups" introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations;Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (ill Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Mean-field models McKean-Vlasov equation Backward stochastic differential equations Comparison theorem dynamic programming principle Viscosity solution

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Tightness and duality of martingale transport on the Skorokhod space

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2017年第3期127卷 927-956页

作者： Guo, Gaoyue Tan, Xiaolu Touzi, Nizar Ecole Polytech CMAP Palaiseau France PSL Res Univ Univ Paris Dauphine CEREMADE Paris France

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle. (C) 2016 Elsevier B.V. All rights reserved.

关键词： S-topology dynamic programming principle Robust superhedging

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UNBOUNDED VISCOSITY SOLUTIONS OF HYBRID CONTROL SYSTEMS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2010年第1期16卷 176-193页

作者： Barles, Guy Dharmatti, Sheetal Ramaswamy, Mythily Univ Tours Lab Math & Phys Theor F-37200 Tours France Univ Toulouse 3 CNRS UMR 5640 Lab MIP F-31062 Toulouse 9 France TIFR Ctr Applicable Math Bangalore 560065 Karnataka India

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.

关键词： dynamic programming principle viscosity solution quasivariational inequality hybrid control

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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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Stochastic control/stopping problem with expectation constraints

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2024年 176卷

作者： Bayraktar, Erhan Yao, Song Univ Michigan Dept Math Ann Arbor MI 48109 USA Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA

We study a stochastic control/stopping problem with a series of inequality-type and equalitytype expectation constraints in a general non-Markovian framework. We demonstrate that the stochastic control/stopping problem with expectation constraints (CSEC) is independent of a specific probability setting and is equivalent to the constrained stochastic control/stopping problem in weak formulation (an optimization over joint laws of Brownian motion, state dynamics, diffusion controls and stopping rules on an enlarged canonical space). Using a martingale-problem formulation of controlled SDEs in spirit of Stroock and Varadhan (2006), we characterize the probability classes in weak formulation by countably many actions of canonical processes, and thus obtain the upper semi-analyticity of the CSEC value function. Then we employ a measurable selection argument to establish a dynamic programming principle (DPP) in weak formulation for the CSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon. This article extends (El Karoui and Tan, 2013) to the expectation-constraint case. We extend our previous work (Bayraktar and Yao, 2024) to the more complicated setting where the diffusion is controlled. Compared to that paper the topological properties of diffusion-control spaces and the corresponding measurability are more technically involved which complicate the arguments especially for the measurable selection for the super-solution side of DPP in the weak formulation.

关键词： Martingale-problem formulation Enlarged canonical space Polish space of diffusion controls Polish space of stopping times dynamic programming principle Regular conditional probability distribution Measurable selection Stochastic control/stopping problem with expectation constraints

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