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

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 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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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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Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic controlled systems*

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2020年第1期26卷 81-81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

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 56 (2018) 4309-4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 57 (2019) 3911-3938] for a fully coupled forward-backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman (HJB) equation combine an algebra equation respectively. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first-order and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622-1662] and Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249-259], and obtain the relationship between the MP from Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249-259] and the DPP from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622-1662].

关键词： Fully coupled forward– backward stochastic differential equations global stochastic maximum principle dynamic programming principle viscosity solution monotonicity condition

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INFINITE TIME HORIZON STOCHASTIC RECURSIVE CONTROL PROBLEMS WITH JUMPS: dynamic programming AND STOCHASTIC VERIFICATION THEOREMS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2025年第2期63卷 796-821页

作者： Luo, Sheng Li, Xun Wei, Qingmeng Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Northeast Normal Univ Sch Math & Stat Changchun 130024 Peoples R China

This paper is devoted to studying an infinite time horizon stochastic recursive control problem with jumps, where an infinite time horizon stochastic differential equation and backward stochastic differential equation with jumps describe the state process and the cost functional, respectively. By establishing the dynamic programming principle, we shed light on the value function of the control problem with an integral-partial differential equation of HJB type in the sense of viscosity solutions. On the other hand, stochastic verification theorems are also studied to provide sufficient conditions to verify the optimality of the given admissible controls. Such a study is carried out within the framework of classical solutions as well as in that of viscosity solutions. Our work emphasizes important differences from the approach for finite time horizon problems. In particular, we have to work in an L-p-setting for p > 4 in order to study the verification theorem in viscosity sense.

关键词： infinite horizon FBSDEs with jumps dynamic programming principle viscosity solution HJB equation integral-differential operators stochastic verification theorem

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The relationship between the maximum principle and dynamic programming for the control of parabolic variational inequalities

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1997年第5期35卷 1711-1738页

作者： Popa, C Facultatea de Matematicǎ Universitatea Al. I. Cuza 6600 Iaşi Bdul. Copou 11 Romania

We show that the well-known relationship between the dual extremal are in the maximum principle and the optimal value function (of dynamic programming), calculated on the optimal trajectory, is valid for the control of parabolic variational inequalities. It follows that every optimal control is given by a feedback law. In the case when the functions defining the performance index are convex also with respect to the state variable, a more specific result is obtained.

关键词： optimal control dual extremal arc optimal value function parabolic variational inequality optimality conditions dynamic programming principle feedback law

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On the relationship between the stochastic maximum principle and dynamic programming in singular stochastic control

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STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES 2012年第2-3期84卷 233-249页

作者： Bahlali, Khaled Chighoub, Farid Mezerdi, Brahim Univ Mohamed Khider Lab Appl Math Mexico City 07000 DF Mexico UTV UFR Sci F-83957 La Garde France

This paper investigates the relationship between the stochastic maximum principle and the dynamic programming principle for singular stochastic control problems. The state of the system under consideration is governed by a stochastic differential equation, with nonlinear coefficients, allowing both classical control and singular control. We show that the necessary conditions for optimality, obtained earlier, are in fact sufficient provided some concavity conditions are fulfilled. In a second step, we prove a verification theorem and we show that the solution of the adjoint equation coincides with the derivative of the value function. Finally, using these results, we solve explicitly an example.

关键词： stochastic maximum principle dynamic programming principle singular control

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dynamic programming principles for Mean-Field Controls with Learning

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OPERATIONS RESEARCH 2023年第4期71卷 1040-1054页

作者： Gu, Haotian Guo, Xin Wei, Xiaoli Xu, Renyuan Univ Calif Berkeley Dept Math Berkeley CA 94720 USA Univ Calif Berkeley Dept Ind Engn & Operat Res Berkeley CA 94720 USA Tsinghua Berkeley Shenzhen Inst Shenzhen 518055 Peoples R China Univ Southern Calif Ind & Syst Engn Los Angeles CA 90001 USA

The dynamic programming principle (DPP) is fundamental for control and optimization, including Markov decision problems (MDPs), reinforcement learning (RL), and, more recently, mean-field controls (MFCs). However, in the learning framework of MFCs, the DPP has not been rigorously established, despite its critical importance for algorithm designs. In this paper, we first present a simple example in MFCs with learning where the DPP fails with a misspecified Q function and then propose the correct form of Q function in an appropriate space for MFCs with learning. This particular form of Q function is different from the classical one and is called the IQ function. In the special case when the transition probability and the reward are independent of the mean-field information, it integrates the classical Q function for single-agent RL over the state-action distribution. In other words, MFCs with learning can be viewed as lifting the classical RLs by replacing the state-action space with its probability distribution space. This identification of the IQ function enables us to establish precisely the DPP in the learning framework of MFCs. Finally, we illustrate through numerical experiments the time consistency of this IQ function.

关键词： mean-field controls dynamic programming principle multi-agent reinforcement learning reinforcement learning Q-learning cooperative game

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Stochastic McKean-Vlasov Control Problem with Regime-Switching and Its Applications

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JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY 2025年 1-25页

作者： Zhang, Liangquan Li, Xun Renmin Univ China Sch Math Beijing 100872 Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong 999077 Peoples R China

This paper focuses on the McKean-Vlasov system's stochastic optimal control problem with Markov regime-switching. To this end, the authors establish a new It & ocirc;'s formula using the linear derivative on the Wasserstein space. This formula enables us to derive the Hamilton-Jacobi-Bellman equation and verification theorems for McKean-Vlasov optimal controls with regime-switching using dynamic programming. As concrete applications, the authors first study the McKean-Vlasov stochastic linear quadratic optimal control problem of the Markov regime-switching system, where all the coefficients can depend on the jump that switches among a finite number of states. Then, the authors represent the optimal control by four highly coupled Riccati equations. Besides, the authors revisit a continuous-time Markowitz mean-variance portfolio selection model (incomplete market) for a market consisting of one bank account and multiple stocks, in which the bank interest rate, the appreciation and volatility rates of the stocks are Markov-modulated. The mean-variance efficient portfolios can be derived explicitly in closed forms based on solutions of four Riccati equations.

关键词： dynamic programming principle Hamilton-Jacobi-Bellman equation McKean-Vlasov regime-switching Wasserstein space value function

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LINEARIZATION TECHNIQUES FOR L^∞-CONTROL PROBLEMS AND dynamic programming principleS IN CLASSICAL AND L^∞-CONTROL PROBLEMS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2012年第3期18卷 836-855页

作者： Goreac, Dan Serea, Oana-Silvia Univ Paris Est Marne la Vallee LAMA UMR8050 F-77454 Champs Sur Marne Marne La Vallee France Ecole Polytech CMAP F-91128 Palaiseau France Univ Perpignan LAMPS F-66860 Perpignan France

The aim of the paper is to provide a linearization approach to the L-infinity-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the L-p approach and the associated linear formulations. This seems to be the most appropriate tool for treating L-infinity problems in continuous and lower semicontinuous setting.

关键词： dynamic programming principle essential supremum HJ equations occupational measures L-p approximations

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