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Relationship Between General MP and DPP for the Stochastic Recursive Optimal Control Problem with Jumps

Journal of Systems Science & Complexity 2024年第6期37卷 2466-2486页

作者： WANG Bin SHI Jingtao School of Mathematics Shandong UniversityJinan 250100China

This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily *** among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,*** examples are given to illustrate the theoretical results.

关键词： Backward stochastic differential equation with jumps dynamic programming principle maximum principle recursive optimal control viscosity solution

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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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Stochastic differential games with controlled regime-switching

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第4期43卷 1-27页

作者： Ma, Chenglin Zhao, Huaizhong Shandong Univ Sch Math Jinan 250100 Peoples R China Univ Durham Dept Math Sci Durham DH1 3LE England Shandong Univ Res Ctr Math & Interdisciplinary Sci Qingdao 266237 Peoples R China

In this article, we consider a two-player zero-sum stochastic differential game with regime-switching. Different from the results in existing literature on stochastic differential games with regime-switching, we consider a game between a Markov chain and a state process which are two fully coupled stochastic processes. The payoff function is given by an integral with random terminal horizon. We first study the continuity of the lower and upper value functions under some additional conditions, based on which we establish the dynamic programming principle. We further prove that the lower and upper value functions are unique viscosity solutions of the associated lower and upper Hamilton-Jacobi-Bellman-Isaacs equations with regime-switching, respectively. These two value functions coincide under the Isaacs condition, which implies that the game admits a value. We finally apply our results to an example.

关键词： Stochastic differential games Controlled regime-switching dynamic programming principle Viscosity solutions Hamilton-Jacobi-Bellman-Isaacs equations

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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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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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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 STOPPING WITH EXPECTATION CONSTRAINTS

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ANNALS OF APPLIED PROBABILITY 2024年第1B期34卷 917-959页

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

We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an arbitrary probability setting is equivalent to the constrained problem in weak formulation (an optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space), and thus the OSEC value is independent of a specific probabilistic setup. Using a martingale-problem formulation, we make an equivalent characterization of the probability classes in weak formulation, which implies that the OSEC value function is upper semianalytic. Then we exploit a measurable selection argument to establish a dynamic programming principle in weak formulation for the OSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon.

关键词： Optimal stopping with expectation constraints martingale-problem formulation enlarged canonical space Polish space of stopping times dynamic programming principle regular conditional probability distribution measurable selection

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Optimal stopping: Bermudan strategies meet non-linear evaluations

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ELECTRONIC JOURNAL OF PROBABILITY 2024年第none期29卷 1-29页

作者： Grigorova, Miryana Quenez, Marie-Claire Yuan, Peng Univ Warwick Coventry England Univ Paris Cite Paris France

We address an optimal stopping problem over the set of Bermudan-type strategies Theta (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators rho . We provide a characterization of the value family V in terms of what we call the (Theta, rho)- Snell envelope of the pay-off family. We establish a dynamic programming principle. We provide an optimality criterion in terms of a (Theta, rho)-martingale property of V on a stochastic interval. We investigate the (Theta, rho)-martingale structure and we show that the "first time" when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario omega . We provide examples of non-linear operators entering our framework.

关键词： optimal stopping Bermudan stopping strategy non-linear operator non-linear evaluation g-expectation dynamic risk measure dynamic programming principle non-linear Snell envelope family Bermudan strategy

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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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