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dynamic programming principle and Hamilton-Jacobi-Bellman Equation under nonlinear expectation

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2022年第0期28卷 25-25页

作者： Hu, Mingshang Ji, Shaolin Li, Xiaojuan Shandong Univ Zhongtai Secur Inst Financial Studies Jinan 250100 Shandong Peoples R China

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under G-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution to a type of fully nonlinear HJB equation.

关键词： dynamic programming principle Hamilton-Jacobi-Bellman equation Stochastic recursive optimal control Backward stochastic differential equation

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On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2013年第8期123卷 3273-3298页

作者： Krylov, N. V. Univ Minnesota Minneapolis MN 55455 USA

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. The zeroth-order "coefficient" and the "free" term are only assumed to be measurable. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. The main assumption, which will be removed in a subsequent article, is that there exists a sufficiently regular solution of the Isaacs equation. (C) 2013 Elsevier B.V. All rights reserved.

关键词： dynamic programming principle Stochastic games Isaacs equation

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Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions

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OPTIMAL CONTROL APPLICATIONS & METHODS 2014年第1期35卷 61-76页

作者： Shi, Jingtao Shandong Univ Sch Math Jinan 250100 Peoples R China

This paper is concerned with the relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions. Under the assumption that the value function is smooth, relations among the adjoint processes, the generalized Hamiltonian function, and the value function are given. A linear quadratic recursive utility portfolio optimization problem in the financial market is discussed to show the applications of the main result. Copyright (c) 2012 John Wiley & Sons, Ltd.

关键词： stochastic optimal control recursive utility backward stochastic differential equation jump diffusions maximum principle dynamic programming principle

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Generalized dynamic programming principle and sparse mean-field control problems

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2020年第1期481卷 123437-123437页

作者： Cavagnari, Giulia Marigonda, Antonio Piccoli, Benedetto Univ Pavia Dept Math F Casorati Via Ferrate 5 I-27100 Pavia Italy Univ Verona Dept Comp Sci Str Le Grazie 15 I-37134 Verona Italy Rutgers Univ Camden Dept Math Sci 311 N 5th St Camden NJ 08102 USA

In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a control sparsity constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract dynamic programming principle, providing further applications in the Appendix. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Multi-agent mean field sparse control Hamilton-Jacobi equation in Wasserstein space Control with uncertainty dynamic programming principle

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On dynamic programming principle for Stochastic Control Under Expectation Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第3期185卷 803-818页

作者： Chow, Yuk-Loong Yu, Xiang Zhou, Chao Sun Yat Sen Univ Sch Math Guangzhou Peoples R China Hong Kong Polytech Univ Dept Appl Math Hung Hom Kowloon Hong Kong Peoples R China Natl Univ Singapore Dept Math Singapore Singapore

This paper studies the dynamic programming principle using the measurable selection method for stochastic control of continuous processes. The novelty of this work is to incorporate intermediate expectation constraints on the canonical space at each time t. Motivated by some financial applications, we show that several types of dynamic trading constraints can be reformulated into expectation constraints on paths of controlled state processes. Our results can therefore be employed to recover the dynamic programming principle for these optimal investment problems under dynamic constraints, possibly path-dependent, in a non-Markovian framework.

关键词： dynamic programming principle Measurable selection Intermediate expectation constraints dynamic trading constraints

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On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

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PROBABILITY THEORY AND RELATED FIELDS 2014年第3-4期158卷 751-783页

作者： Krylov, N. V. Univ Minnesota Minneapolis MN 55455 USA

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order "coefficient" and the "free" term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.

关键词： dynamic programming principle Stochastic games Isaacs equations

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Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2020年第1期46卷 1-28页

作者： Alla, Alessandro Haasdonk, Bernard Schmidt, Andreas Pontificia Univ Catolica Rio de Janeiro Dept Math Rua Marques de Sao Vicente 225 BR-22453900 Rio de Janeiro Brazil Univ Stuttgart Inst Appl Anal & Numer Simulat Pfaffenwaldring 57 D-70569 Stuttgart Germany

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.

关键词： dynamic programming principle Semi-Lagrangian schemes Hamilton-Jacobi-Bellman equations Optimal control Model reduction Reduced basis method

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dynamic programming principle for Classical and Singular Stochastic Control with Discretionary Stopping

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APPLIED MATHEMATICS AND OPTIMIZATION 2023年第1期88卷 7-7页

作者： De Angelis, Tiziano Milazzo, Alessandro Univ Torino Sch Management & Econ Dept ESOMAS Cso Unione Sovietica 218bis I-10134 Turin Italy Coll Carlo Alberto Pza Arbarello 8 I-10122 Turin Italy Uppsala Univ Dept Math Box 480 S-75106 Uppsala Sweden

We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a d-dimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We consider both the cases in which the total available fuel for the singular control is either bounded or unbounded. We build upon existing proofs of DPP and extend results available in the traditional literature on singular control (Haussmann and Suo in SIAM J Control Optim 33(3):916-936, 1995;SIAM J Control Optim 33(3):937-959, 1995) by relaxing some key assumptions and including the discretionary stopping feature. We also connect with more general versions of the DPP (e.g., Bouchard and Touzi in SIAM J Control Optim 49(3):948-962, 2011) by showing in detail how our class of problems meets the abstract requirements therein.

关键词： dynamic programming principle Stochastic control Singular control Discretionary stopping

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RANDOMIZED dynamic programming principle AND FEYNMAN-KAC REPRESENTATION FOR OPTIMAL CONTROL OF MCKEAN-VLASOV dynamicS

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2018年第3期370卷 2115-2160页

作者： Bayraktar, Erhan Cosso, Andrea Pham, Huyen Univ Michigan Dept Math 530 Church St Ann Arbor MI 48109 USA Politecn Milan Dipartimento Matemat Via Bonardi 9 I-20133 Milan Italy Univ Bologna Dipartimento Matemat Piazza Porta S Donato 5 I-40126 Bologna Italy Univ Paris Diderot Lab Probabilites & Modeles Aleatoires CNRS UMR 7599 F-75205 Paris 13 France Univ Paris Diderot Lab Probabilites & Modeles Aleatoires CNRS UMR 7599 Crest France

We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V admits a nonlinear FeynmanKac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V. The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V. The DPP is important in obtaining a characterization of the value function as a solution of a nonlinear partial differential equation (the so-called HamiltonJacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the FeynmanKac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists of formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V of the original control problem.

关键词： Controlled McKean-Vlasov stochastic differential equations dynamic programming principle randomization method forward-backward stochastic differential equations

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MCKEAN-VLASOV OPTIMAL CONTROL: THE dynamic programming principle

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ANNALS OF PROBABILITY 2022年第2期50卷 791-833页

作者： Djete, Mao Fabrice Possamai, Dylan Tan, Xiaolu Ecole Polytech Paris France Swiss Fed Inst Technol Dept Math Zurich Switzerland Chinese Univ Hong Kong Dept Math Hong Kong Peoples R China

We study the McKean-Vlasov optimal control problem with common noise which allow the law of the control process to appear in the state dynamics under various formulations: strong and weak ones, Markovian or non-Markovian. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.

关键词： McKean-Vlasov optimal control dynamic programming principle measurable selection

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