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dynamic programming principle for stochastic optimal control problem under degenerate G-expectation

SYSTEMS & CONTROL LETTERS 2025年 196卷

作者： Li, Xiaojuan Qilu Normal Univ Dept Math Jinan 250200 Peoples R China

In this paper, we study a stochastic optimal control problem under degenerate G-expectation. By using implied partition method, we show that the approximation result for admissible controls still hold. Based on this result, we obtain the dynamic programming principle, and prove that the value function is the unique viscosity solution to the related HJB equation under degenerate case.

关键词： G-expectation dynamic programming principle Hamilton-Jacobi-Bellman equation Stochastic optimal control

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Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problem under volatility uncertainty

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OPTIMAL CONTROL APPLICATIONS & METHODS 2023年第5期44卷 2457-2475页

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

In this article, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for stochastic recursive optimal control problem driven by G-Brownian motion. Under the smooth assumption for the value function, we obtain the connection between MP and DPP under a reference probability P-t,x(*). Within the framework of viscosity solution, we establish the relation between the first-order super-jet, sub-jet of the value function and the solution to the adjoint equation respectively.

关键词： G-expectation backward stochastic differential equation dynamic programming principle maximum principle stochastic recursive optimal control

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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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dynamic programming principle and computable prices in financial market models with transaction costs

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2023年第2期524卷

作者： Lepinette, Emmanuel Vu, Duc Thinh Paris Dauphine Univ Ceremade CNRS PSL Natl ResUMR 7534 Pl Marechal De Lattre Tassigny F-75775 Paris 16 France Fac Sci Tunis Gosaef Tunis Tunisia

How to compute (super) hedging costs in rather general financial market models with transaction costs in discrete-time? Despite the huge literature on this topic, most of results are characterizations of the super-hedging prices while it remains difficult to deduce numerical procedure to estimate them. We establish here a dynamic programming principle and we prove that it is possible to implement it under some conditions on the conditional supports of the price and volume processes for a large class of market models including convex costs such as order books but also non convex costs, e.g. fixed cost models.(c) 2023 Elsevier Inc. All rights reserved.

关键词： Hedging costs European options dynamic programming principle AIP condition Random set theory No-arbitrage condition

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

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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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dynamic programming principle for backward doubly stochastic recursive optimal control problem and sobolev weak solution of the stochastic Hamilton-Jacobi-Bellman equation

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Fundamental Research 2024年

作者： Li, Yunhong Matoussi, Anis Wei, Lifeng Wu, Zhen Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong Risk and Insurance Institute of Le Mans Laboratoire Manceau de Mathématiques Le Mans University Sarthe Le Mans 72000 France School of Mathematical Sciences Ocean University of China Shandong Qingdao 266003 China Laboratory of Marine Mathematics Ocean University of China Shandong Qingdao 266003 China School of Mathematics Shandong University Shandong Jinan 250100 China

In this paper, we investigate a backward doubly stochastic recursive optimal control problem wherein the cost function is expressed as the solution to a backward doubly stochastic differential equation. We present the dynamical programming principle for this type of optimal control problem and establish that the value function is the unique Sobolev weak solution to the associated stochastic Hamilton-Jacobi-Bellman equation. © 2023

关键词： Backward doubly stochastic differential equation dynamic programming principle Hamilton-Jacobi-Bellman equation Recursive optimal control Sobolev weak solution

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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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dynamic programming principle AND HAMILTON-JACOBI-BELLMAN EQUATIONS FOR FRACTIONAL-ORDER SYSTEMS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2020年第6期58卷 3185-3211页

作者： Gomoyunov, Mikhail, I Russian Acad Sci Ural Branch NN Krasovskii Inst Math & Mech Ekaterinburg 620108 Russia Ural Fed Univ Ekaterinburg 620002 Russia

We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order alpha is an element of (0, 1). The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order alpha, we associate the considered optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.

关键词： optimal control fractional derivatives dynamic programming principle Hamilton-Jacobi-Bellman equation coinvariant derivatives feedback control

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The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2024年 30卷

作者： Dong, Yuchao Meng, Qingxin Zhang, Qi Tongji Univ Sch Math Sci Key Lab Intelligent Comp & Applicat Minist Educ Shanghai 200092 Peoples R China Huzhou Univ Dept Math Sci Huzhou 313000 Zhejiang Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Fudan Univ Lab Math Nonlinear Sci Shanghai 200433 Peoples R China

This paper aims to explore the relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients. Under certain regular conditions for the coefficients, the relationship between the Hamiltonian system with random coefficients and stochastic Hamilton-Jacobi-Bellman equation is obtained. It is very different from the deterministic coefficients case since stochastic Hamilton-Jacobi-Bellman equation is a backward stochastic partial differential equation with solution being a pair of random fields rather than a deterministic function. A linear quadratic recursive optimization problem is given as an explicitly illustrated example based on this kind of relationship.

关键词： Maximum principle dynamic programming principle random coefficient stochastic recursive control backward stochastic partial differential equation

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A Framework for the dynamic programming principle and Martingale-Generated Control Correspondences

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APPLIED MATHEMATICS AND OPTIMIZATION 2021年第3期83卷 1311-1352页

作者： Fayvisovich, Roman Zitkovic, Gordan Univ Texas Austin Dept Math Austin TX 78712 USA

We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the "martingale formulation" with particular ease. We give two illustrations;first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton-Jacobi-Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem.

关键词： dynamic programming principle Optimal stochastic control Control correspondences Martingale problem Singular control

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