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Stochastic Differential Games of Mean-Field dynamics and Second-Order Bellman–Isaacs Equations on the Wasserstein Space

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Acta Mathematica Sinica,English Series 2025年第3期41卷 873-907页

作者： Tao Hao Jie Xiong School of Statistics and Mathematics Shandong University of Finance and EconomicsJi’nan250014P.R.China Department of Mathematics and SUSTech International Center for Mathematics Southern University of Science and TechnologyShenzhen518055P.R.China

This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the *** our game,both the(lower and upper)value functions and the(lower and upper)second-order Bellman–Isaacs equations are defined on the Wasserstein space P_(2)(R^(n))which is an infinite dimensional *** dynamic programming principle for the value functions is *** the(upper and lower)value functions are smooth enough,we show that they are the classical solutions to the second-order Bellman–Isaacs *** the other hand,the classical solutions to the(upper and lower)Bellman–Isaacs equations are unique and coincide with the(upper and lower)value *** an illustrative application,the linear quadratic case is *** the Isaacs condition,the explicit expressions of optimal closed-loop controls for both players are ***,we introduce the intrinsic notion of viscosity solution of our second-order Bellman–Isaacs equations,and characterize the(upper and lower)value functions as their viscosity solutions.

关键词： Stochastic differential game Wasserstein space viscosity solution dynamic programming principle mean-field stochastic differential equation Riccati equation

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Tug-of-War Games Related to Oblique Derivative Boundary Value Problems with the Normalized p-Laplacian

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POTENTIAL ANALYSIS 2025年 1-35页

作者： Han, Jeongmin Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland Pusan Natl Univ Dept Math Pusan 46241 South Korea

In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem Delta pNu=0in Omega,+gamma u=gamma Gon partial derivative Omega,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _{p}<^>{N}u=0 & \text {in} \,\,\, \Omega ,\\ \langle \beta , Du \rangle + \gamma u = \gamma G & \text {on} \,\,\, \partial \Omega ,\\ \end{array} \right. \end{aligned}$$\end{document}where Delta pN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{p}<^>{N}$$\end{document} is the normalized p-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using 'shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence.

关键词： dynamic programming principle Stochastic game Oblique derivative boundary condition Viscosity solution

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Controlled superprocesses and HJB equation in the space of finite measures

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2025年第2期547卷

作者： Ocello, Antonio Sorbonne Univ LPSM UMR CNRS 8001 4 Pl Jussieu F-75005 Paris France Univ Paris Cite 4 Pl Jussieu F-75005 Paris France

This paper introduces the formalism required to analyze a certain class of stochastic control problems that involve a super diffusion as the underlying controlled system. To establish the existence of these processes, we show that they are weak scaling limits of controlled branching processes. First, we prove a generalized It & ocirc;'s formula for this dynamics in the space of finite measures, using the differentiation in the space of finite positive measures. This lays the groundwork for a PDE characterization of the value function of a control problem, which leads to a verification theorem. Finally, focusing on an exponential-type value function, we show how a regular solution to a finite-dimensional HJB equation can be used to construct a smooth solution to the HJB equation in the space of finite measures, via the so-called branching property technique. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： Stochastic control Superprocesses Hamilton-Jacobi-Bellman equation dynamic programming principle

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Optimal Covid-19 Control on Effectiveness of Detection Campaign and Treatment

INTERNATIONAL JOURNAL OF MATHEMATICAL ENGINEERING AND MANAGE...

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INTERNATIONAL JOURNAL OF MATHEMATICAL ENGINEERING AND MANAGEMENT SCIENCES 2025年第2期10卷 420-440页

作者： Balde, Mouhamadou A. M. T. Ly, Sidy Tendeng, Lena Univ Cheikh Anta Diop Lab Math Decis & Numer Anal BP 5005 Dakar Senegal

The COVID-19 pandemic has seen the development of several mathematical models. In recent years, the very topical issue of re- susceptibility has led to the proposal of more complex models to address this issue. The paper deals with an optimal control problem applied to COVID-19. The Pontryagin maximum principle and the dynamic programming principle are used to solve the problem. A compartmental Ordinary Differential Equation (ODE) model is proposed to study the evolution of the pandemic by controlling the effectiveness of the detection campaign and the treatment. We prove the global stability of the Disease-Free Equilibrium (DFE) and the existence of optimal control and trajectories of the model. In the optimal control problem, we bring the system back to the DFE. Numerical simulations based on COVID-19 data in Senegal show possibilities to reduce the disease evolution, sometimes by emphasizing the detection campaign and/or the treatment proposed to patients.

关键词： Differential equations dynamic programming principle Hamilton-Jacobi-Bellman equations Pontryagin maximum principle Covid-19

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Representation Results and Error Estimates for Differential Games with Applications Using Neural Networks

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dynamic GAMES AND APPLICATIONS 2025年第2期15卷 417-453页

作者： Bokanowski, Olivier Warin, Xavier Univ Paris Cite Lab Jacques Louis F-75013 Paris France Sorbonne Univ CNRS LJLL F-75005 Paris France EDF R&D & FiME F-91120 Palaiseau France

We study deterministic optimal control problems for differential games with finite horizon. We propose new approximations of the strategies in feedback form and show error estimates and a convergence result of the value in some weak sense for one of the formulations. This result applies in particular to neural network approximations. This work follows some ideas introduced in Bokanowski, Prost and Warin (PDEA, 2023) for deterministic optimal control problems, yet with a simplified approach for the error estimates, which allows to consider a global optimization scheme instead of a time-marching scheme. We also give a new approximation result between the continuous and the semi-discrete optimal control value in the game setting, improving the classical convergence order O(Delta t1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({\Delta t}<^>{1/2})$$\end{document} to O(Delta t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({\Delta t})$$\end{document}, under some assumptions on the dynamical system. Numerical examples are performed on elementary academic problems related to backward reachability, with exact analytic solutions given, as well as a two-player game in the presence of state constraints, using stochastic gradient-type algorithms to deal with the minimax problem.

关键词： Differential games Two-player games Neural networks Deterministic optimal control dynamic programming principle Hamilton Jacobi Isaacs equation Front propagation Level sets Non-anticipative strategies

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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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dynamic programming for Finite Ensembles of Nanomagnetic Particles

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JOURNAL OF SCIENTIFIC COMPUTING 2019年第1期80卷 351-375页

作者： Jensen, Max Majee, Ananta K. Prohl, Andreas Schellnegger, Christian Univ Sussex Dept Math Pevensey 2 BldgFalmer Campus Brighton BN1 9QH E Sussex England Indian Inst Technol Delhi Dept Math Hauz Khas New Delhi 110016 India Univ Tubingen Math Inst Morgenstelle 10 D-72076 Tubingen Germany

We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. Using the dynamic programming principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf-Cole transformation, the associated Hamilton-Jacobi-Bellman equation of the dynamic programming principle may be re-cast into a linear PDE on the manifold M=(S2)N, whose classical solution may be represented via Feynman-Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics.

关键词： Stochastic Landau-Lifschitz-Gilbert equation Stratonovich noise HJB equation dynamic programming principle Hopf-Cole transformation Discretization

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A note on negative dynamic programming for risk-sensitive control

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OPERATIONS RESEARCH LETTERS 2008年第5期36卷 531-534页

作者： Jaskiewicz, Anna Polish Acad Sci Inst Math PL-00956 Warsaw Poland

Negative dynamic programming for risk-sensitive control is studied. Under some compactness and semicontinuity assumptions the following results are proved: the convergence of the value iteration algorithm to the optimal expected total reward, the Borel measurability or upper semicontinuity of the optimal value functions, and the existence of an optimal stationary policy. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Risk-sensitive control Borel state space dynamic programming principle

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SINGULAR OPTIMAL STOCHASTIC CONTROLS .2. dynamic-programming

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1995年第3期33卷 937-959页

作者： HAUSSMANN, UG SUO, WL UNIV TORONTO FAC MANAGEMENTTORONTOON M5S 1V4CANADA

The dynamic programming principle for a multidimensional singular stochastic control problem is established in this paper. When assuming Lipschitz continuity on the data, it is shown that the value function is continu... 详细信息

关键词： SINGULAR CONTROLS CONTROL RULES VALUE FUNCTION dynamic programming principle HAMILTON-JACOBI-BELLMAN EQUATION VISCOSITY SOLUTION

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dynamic programming FOR OPTIMAL CONTROL OF STOCHASTIC MCKEAN-VLASOV dynamicS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第2期55卷 1069-1101页

作者： Pham, Huyen Wei, Xiaoli CNRS UMR 7599 Lab Probabilites & Modele Aleatoires Paris France Univ Paris Diderot Paris France CREST ENSAE Paris France Univ Paris Diderot CNRS Lab Probabilites & Modeles Aleatoires UMR 7599 Paris France

We study optimal control of the general stochastic McKean-Vlasov equation. Such a problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to [P.L. Lions, Cours au College de France : Theorie des jeux a champ moyens, (2012), pp. 2006-2012] and Ito's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.

关键词： stochastic McKean-Vlasov SDEs dynamic programming principle Bellman equation Wasserstein space viscosity solutions

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