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Non-asymptotic Convergence Rates for Mean-Field Games: Weak Formulation and McKean–Vlasov BSDEs 平均场对策的非渐近收敛率：弱公式和McKean-Vlasov BSDEs
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-11 DOI: 10.1007/s00245-025-10256-4
Dylan Possamaï, Ludovic Tangpi
{"title":"Non-asymptotic Convergence Rates for Mean-Field Games: Weak Formulation and McKean–Vlasov BSDEs","authors":"Dylan Possamaï,&nbsp;Ludovic Tangpi","doi":"10.1007/s00245-025-10256-4","DOIUrl":"10.1007/s00245-025-10256-4","url":null,"abstract":"<div><p>This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker (Ann Appl Probab 25(3):1189–1231, 2015), we consider a fully non-Markovian setting allowing for drift control and interactions through the joint distribution of players’ states and controls. We provide first a characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean–Vlasov backward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short-time horizon, separability assumptions on the coefficients, nor Lasry and Lions’s monotonicity conditions, but rather smallness—or alternatively regularity—conditions on the terminal reward and a dissipativity condition on the drift. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the <i>N</i>-player version to their mean-field counterparts, for general open-loop equilibria. An appropriate reformulation of our approach also allows us to treat closed-loop equilibria, and to obtain convergence results for the master equation associated to the problem.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10256-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143818246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Invariant Sample Measures for Non-autonomous Stochastic Non-local Discrete p-Laplacian Equations with Time Varying Delay and Multiplicative Noises 具有时变延迟和乘法噪声的非自治随机非局部离散 p-Laplacian 方程的不变抽样量纲
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-05 DOI: 10.1007/s00245-025-10254-6
Wenqiang Zhao, Xia Liu
{"title":"Invariant Sample Measures for Non-autonomous Stochastic Non-local Discrete p-Laplacian Equations with Time Varying Delay and Multiplicative Noises","authors":"Wenqiang Zhao,&nbsp;Xia Liu","doi":"10.1007/s00245-025-10254-6","DOIUrl":"10.1007/s00245-025-10254-6","url":null,"abstract":"<div><p>In this article, a class of <i>non-local</i> stochastic lattice models with fractional powers of the discrete <i>p</i>-Laplacian, incorporating time-varying-distributed delay as well as multiplicative noises at each node, are propounded. First, by utilizing an existence theorem of solutions for infinite ordinary differential equations, we prove the well-posedness of the stochastic equations, whose solution operator admits an NRDS. Besides, the tempered random attractor is constructed for this NRDS. Finally, with the identical conditions, we prove that the solution is jointly continuous in initial time and initial data, and eventually, a family of invariant sample Borel probability measures supported in the random attractors are established in a Banach space (not a Hilbert space) for the corresponding NRDS.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143778001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Analysis of Subdifferentials of Marginal and Performance Functions 边际函数和性能函数的次微分分析
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-02 DOI: 10.1007/s00245-025-10251-9
Duong Thi Viet An, Jean-Paul Penot
{"title":"Analysis of Subdifferentials of Marginal and Performance Functions","authors":"Duong Thi Viet An,&nbsp;Jean-Paul Penot","doi":"10.1007/s00245-025-10251-9","DOIUrl":"10.1007/s00245-025-10251-9","url":null,"abstract":"<div><p>We study generalized derivatives of value functions for optimization problems depending on a parameter <i>w</i>. Interpretations of the results obtained with these substitutes to derivatives are known to be important. We endeavour to answer the question: can one obtain these results without knowing the nature of these substitutes and their constructions? Is there a means to obtain them in a unified way?</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Well-Posedness and Long Time Dynamics for a Coupled Nonlinear Bridge System with Past History 论具有过去历史的耦合非线性桥梁系统的拟合优度和长时动力学
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-02 DOI: 10.1007/s00245-025-10252-8
Soh Edwin Mukiawa, Salim A. Messaoudi
{"title":"On the Well-Posedness and Long Time Dynamics for a Coupled Nonlinear Bridge System with Past History","authors":"Soh Edwin Mukiawa,&nbsp;Salim A. Messaoudi","doi":"10.1007/s00245-025-10252-8","DOIUrl":"10.1007/s00245-025-10252-8","url":null,"abstract":"<div><p>This work is concerned with a coupled nonlinear mathematical model for a suspension bridge with past history. The vibrations of both the road bed in the vertical plain and main cable from which the road bed is suspended by the tie cables are taken into consideration. Using the semi-group approach, we give a thorough and careful existence and uniqueness result. Also, we prove that the associated solution semi-group has a compact global attractor in an appropriate Hilbert space.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Well-Posedness of 3D MHD Equations with Damping on Time-Varying Domains 时变域上具有阻尼的三维多流体力学方程的良好拟合
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-02 DOI: 10.1007/s00245-025-10253-7
Xiaoya Song
{"title":"Well-Posedness of 3D MHD Equations with Damping on Time-Varying Domains","authors":"Xiaoya Song","doi":"10.1007/s00245-025-10253-7","DOIUrl":"10.1007/s00245-025-10253-7","url":null,"abstract":"<div><p>In the present paper, we consider the well-posedness of 3D MHD equations with the damping terms <span>(|u|^{alpha -1}u)</span> and <span>(|B|^{beta -1}B)</span> (<span>(alpha ,beta ge 1)</span>) defined on time-varying domains with homogeneous Dirichlet boundary conditions. We show that the damped 3D MHD system has global weak solutions for any <span>(1le alpha ,beta le 5)</span> and the weak solution is unique for any <span>(4le alpha ,beta le 5)</span>.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved Description of Blaschke–Santaló Diagrams via Numerical Shape Optimization 通过数值形状优化改进对布拉什克-桑塔洛图的描述
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-02 DOI: 10.1007/s00245-025-10250-w
Ilias Ftouhi
{"title":"Improved Description of Blaschke–Santaló Diagrams via Numerical Shape Optimization","authors":"Ilias Ftouhi","doi":"10.1007/s00245-025-10250-w","DOIUrl":"10.1007/s00245-025-10250-w","url":null,"abstract":"<div><p>We propose a method based on the combination of theoretical results on Blaschke–Santaló diagrams and numerical shape optimization techniques to obtain improved description of Blaschke–Santaló diagrams in the class of planar convex sets. To illustrate our approach, we study three relevant diagrams involving the perimeter <i>P</i>, the diameter <i>d</i>, the area <i>A</i> and the first eigenvalue of the Laplace operator with Dirichlet boundary condition <span>(lambda _1)</span>. The first diagram is a purely geometric one involving the triplet (<i>P</i>, <i>d</i>, <i>A</i>) and the two other diagrams involve geometric and spectral functionals, namely <span>((P,lambda _1,A))</span> and <span>((d,lambda _1,A))</span> where a strange phenomenon of non-continuity of the extremal shapes is observed.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10250-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
General Duality and Dual Attainment for Adapted Transport 适应性运输的一般双重性和双重实现
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-03-21 DOI: 10.1007/s00245-025-10240-y
Daniel Kršek, Gudmund Pammer
{"title":"General Duality and Dual Attainment for Adapted Transport","authors":"Daniel Kršek,&nbsp;Gudmund Pammer","doi":"10.1007/s00245-025-10240-y","DOIUrl":"10.1007/s00245-025-10240-y","url":null,"abstract":"<div><p>We investigate duality and existence of dual optimizers for several adapted optimal transport problems under minimal assumptions. This includes the causal and bicausal transport, the causal and bicausal barycenter problem, and a multimarginal problem incorporating causality constraints. Moreover, we characterize polar sets in the causal and bicausal setting and discuss applications of our results in robust finance. We consider a non-dominated model of several financial markets where stocks are traded dynamically, but the joint stock dynamics are unknown. We show that a no-arbitrage assumption naturally leads to sets of multicausal couplings. Consequently, computing the robust superhedging price is equivalent to solving an adapted transport problem, and finding a superhedging strategy means solving the corresponding dual.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10240-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lévy Driven Stochastic Heat Equation with Logarithmic Nonlinearity: Well-Posedness and Large Deviation Principle 具有对数非线性的lsamvy驱动随机热方程：适定性和大偏差原理
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-03-19 DOI: 10.1007/s00245-025-10247-5
R. Kavin, Ananta K. Majee
{"title":"Lévy Driven Stochastic Heat Equation with Logarithmic Nonlinearity: Well-Posedness and Large Deviation Principle","authors":"R. Kavin,&nbsp;Ananta K. Majee","doi":"10.1007/s00245-025-10247-5","DOIUrl":"10.1007/s00245-025-10247-5","url":null,"abstract":"<div><p>In this article, we study the well-posedness theory for solutions of the stochastic heat equations with logarithmic nonlinearity perturbed by multiplicative Lévy noise. By using Aldous tightness criteria and Jakubowski’s version of the Skorokhod theorem on non-metric spaces along with the standard <span>(L^2)</span>-method, we establish the existence of a path-wise unique strong solution. Moreover, by using a weak convergence method, we establish a large deviation principle for the strong solution of the underlying problem. Due to the lack of linear growth and locally Lipschitzness of the term <span>( u log (|u|))</span> present in the underlying problem, the logarithmic Sobolev inequality and the nonlinear versions of Gronwall’s inequalities play a crucial role.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143645648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Policy Iteration for Exploratory Hamilton–Jacobi–Bellman Equations 探索性Hamilton-Jacobi-Bellman方程的策略迭代
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-03-17 DOI: 10.1007/s00245-025-10249-3
Hung Vinh Tran, Zhenhua Wang, Yuming Paul Zhang
{"title":"Policy Iteration for Exploratory Hamilton–Jacobi–Bellman Equations","authors":"Hung Vinh Tran,&nbsp;Zhenhua Wang,&nbsp;Yuming Paul Zhang","doi":"10.1007/s00245-025-10249-3","DOIUrl":"10.1007/s00245-025-10249-3","url":null,"abstract":"<div><p>We study the policy iteration algorithm (PIA) for entropy-regularized stochastic control problems on an infinite time horizon with a large discount rate, focusing on two main scenarios. First, we analyze PIA with bounded coefficients where the controls applied to the diffusion term satisfy a smallness condition. We demonstrate the convergence of PIA based on a uniform <span>({{mathcal {C}}}^{2,alpha })</span> estimate for the value sequence generated by PIA, and provide a quantitative convergence analysis for this scenario. Second, we investigate PIA with unbounded coefficients but no control over the diffusion term. In this scenario, we first provide the well-posedness of the exploratory Hamilton–Jacobi–Bellman equation with linear growth coefficients and polynomial growth reward function. By such a well-posedess result we achieve PIA’s convergence by establishing a quantitative locally uniform <span>({{mathcal {C}}}^{1,alpha })</span> estimates for the generated value sequence.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143632500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Delay Nonlocal Quasilinear Chafee–Infante Problem: An Approach via Semigroup Theory 一类时滞非局部拟线性Chafee-Infante问题：基于半群理论的方法
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-03-10 DOI: 10.1007/s00245-025-10241-x
Tomás Caraballo, A. N. Carvalho, Yessica Julio
{"title":"A Delay Nonlocal Quasilinear Chafee–Infante Problem: An Approach via Semigroup Theory","authors":"Tomás Caraballo,&nbsp;A. N. Carvalho,&nbsp;Yessica Julio","doi":"10.1007/s00245-025-10241-x","DOIUrl":"10.1007/s00245-025-10241-x","url":null,"abstract":"<div><p>In this work we study a dissipative one dimensional scalar parabolic problem with non-local nonlinear diffusion with delay. We consider the general situation in which the functions involved are only continuous and solutions may not be unique. We establish conditions for global existence and prove the existence of global attractors. All results are presented only in the autonomous since the non-autonomous case follows in the same way, including the existence of pullback attractors. A particularly interesting feature is that there is a semilinear problem (nonlocal in space and in time) from which one can obtain all solutions of the associated quasilinear problem and that for this semilinear problem the delay depends on the initial function making its study more involved.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143583402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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