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Stochastic recursive optimal control of McKean-Vlasov type: A viscosity solution approach

JOURNAL OF DIFFERENTIAL EQUATIONS 2024年 409卷 334-394页

作者： Zhang, Liangquan Renmin Univ China Sch Math Beijing 100872 Peoples R China

In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean-Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the infinite dimensional Hamilton-Jacobi-Bellman (HJB for short) equation derived from the decoupling field of the FBSDEs posed by Carmona and Delarue (2015) [28]. At the beginning, the value function is defined by the solution to the controlled BSDE alluded to earlier. On one hand, we can prove the value function is deterministic function with respect to the initial random variable;On the other hand, we can show that the value function is law-invariant, i.e., depending on only via its distribution by virtue of BSDE property. Afterward, utilizing the notion of differentiability with respect to probability measures introduced by P.L. Lions [50], we are able to establish a DPP for the value function in the Wasserstein space of probability measures based on the application of BSDE approach, particularly, employing the notion of stochastic backward semigroups associated with stochastic optimal control problems and It & ocirc;formula along a flow property of the conditional law of the controlled forward state process. We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space. Finally, as an application, we formulate an optimal control problem for linear stochastic differential equations with quadratic cost functionals of McKean-Vlasov type under nonlinear expectation, g-expectation introduced by Peng [43] and derive the optimal feedback control explicitly by means of several groups of Riccati equations. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： dynamic programming principle Forward-backward McKean-Vlasov stochastic differential equations Hamilton-Jacobi-Bellman equation Wasserstein space Value function Viscosity solutions

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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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Holder estimate for a tug-of-war game with 1 <p <2 from Krylov-Safonov regularity theory

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REVISTA MATEMATICA IBEROAMERICANA 2024年第3期40卷 1023-1044页

作者： Arroyo, Angel Parviainen, Mikko Univ Alicante Dept Matemat Alicante 03690 Spain Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the p -Laplacian with 1 < p < 2 . For this version, the asymptotic Holder continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.

关键词： ABP-estimate elliptic non-divergence form partial differential equation with bounded and measurable coefficients dynamic programming principle local H & ouml lder estimate p-Laplacian Pucci extremal operator tug-of-war with noise

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A Zero-Sum Deterministic Impulse Controls Game in Infinite Horizon with a New HJBI-QVI

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APPLIED MATHEMATICS AND OPTIMIZATION 2023年第3期88卷 1-31页

作者： El Asri, Brahim Lalioui, Hafid Mazid, Sehail Ibn Zohr Univ Lab LISAD ENSA Equipe Aide Decis BP 1136 Agadir Morocco Ibn Zohr Univ Fac Sci Dept Math Agadir Morocco

In the present paper, we study a two-player, zero-sum, deterministic differential game with both players adopting impulse controls in infinite-time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI-QVI for which, under a proportional property assumption on the maximizing player cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide.

关键词： Deterministic differential game Impulse control Infinite horizon dynamic programming principle Viscosity solution Quasi-variational inequality

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OPTIMAL INVESTMENT STRATEGY FOR AN INSURER WITH PARTIAL INFORMATION IN CAPITAL AND INSURANCE MARKETS

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第7期19卷 5249-5271页

作者： Xie, Lin Li, Danping Qian, Linyi Chen, Lv Yang, Zhixin East China Normal Univ Sch Stat Key Lab Adv Theory & Applicat Stat & Data Sci MOE Shanghai 200062 Peoples R China East China Normal Univ Acad Stat & Interdisciplinary Sci Key Lab Adv Theory & Applicat Stat & Data Sci MOE Shanghai 200062 Peoples R China Ball State Univ Dept Math Sci Muncie IN 47304 USA

This paper considers a framework in which an insurer determines the optimal investment strategies to maximize the expected utility of terminal wealth. We obtain the optimal investment strategies assuming that both the capital market and the insurance market are partially observable. By employing Bayesian method and filtering theory, we first transform the optimization problem with partial information into the one with complete information. We then achieve the explicit expression of the optimal investment strategy by using dynamic programming principle. In addition, we also derive the optimal investment strategies with complete information in both markets as well as partial information in either market. Finally, we compare the optimal strategies in different models and study value functions numerically to illustrate our results.

关键词： Optimal investment strategy partial information dynamic programming principle filtering theory bayesian approach

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OPTIMAL CONTRACTS AND ASSET PRICES IN A CONTINUOUS-TIME DELEGATED PORTFOLIO MANAGEMENT PROBLEM

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第5期19卷 3186-3216页

作者： Dou, Zheng Lai, Shaoyong Xihua Univ Sch Econ Chengdu 611139 Peoples R China Southwestern Univ Finance & Econ Sch Econ Math Chengdu 611130 Peoples R China

We study optimal contracts and asset prices in a financial market in which an investor delegates a portfolio manager to manage her wealth. The agency frictions are caused by the manager's "shirking" action and hidden effort. The shirking action converts part of the return of the managed portfolio into the manager's income without reducing his utility. The manager's effort improves the return of the portfolio but reduces the manager's utility. We illustrate this dynamic principal-agent problem under hidden effort and observable effort, respectively. When the effort is hidden, to alleviate the impact of moral hazard, the investor pays more for the manager's performance and always keeps the optimal contract related to the returns of the manager's portfolio and market portfolio, and their quadratic (co)variations. When the manager's effort is observable, the optimal contract is related to the return of the market portfolio if the agency friction caused by the shirking action is serious, but is only related to the return of the manager's portfolio if shirking is not serious. Analysing the expected utility of the manager, we find that he has a disposition to hide information about effort to pursue a higher expected utility.

关键词： Optimal contracts asset-management dynamic programming principle principal-agent problem moral hazard

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The Robin Mean Value Equation I: A Random Walk Approach to the Third Boundary Value Problem

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POTENTIAL ANALYSIS 2023年第4期59卷 1695-1726页

作者： Lewicka, Marta Peres, Yuval Univ Pittsburgh Dept Math 139 Univ Pl Pittsburgh PA 15260 USA

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C-1,C-1 -regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W-2,W- P solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks.

关键词： Robin problem Third boundary value problem Oblique boundary conditions dynamic programming principle Random walk Finite difference approximations Viscosity solutions

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The Robin Mean Value Equation II: Asymptotic Holder Regularity

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POTENTIAL ANALYSIS 2023年第4期59卷 2035-2069页

作者： Lewicka, Marta Peres, Yuval Univ Pittsburgh Dept Math 139 Univ Pl Pittsburgh PA 15260 USA

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any C-1,C-1-regular domain and with any bounded Borel right hand side. When compared with the case of continuous right hand side, analyzed in Lewicka and Peres (2022), the present more general setting presents significant technical challenges. Along the way, we prove the asymptotic Holder equicontinuity of solutions to (RMV): Lipschitz in the interior and C-0,C-alpha up to the boundary, for any alpha is an element of (0, 1). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle for a discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks.

关键词： Robin problem Third boundary value problem Oblique boundary conditions dynamic programming principle Random walks Finite difference approximations Viscosity solutions

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Game-Theoretic Approach to Holder Regularity for PDEs Involving Eigenvalues of the Hessian

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POTENTIAL ANALYSIS 2023年第4期59卷 1995-2015页

作者： Blanc, Pablo Han, Jeongmin Parviainen, Mikko Ruosteenoja, Eero Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We prove a local Holder estimate for any exponent 0 0 and alpha(2),..., alpha(n-1) >= 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for visco... 详细信息

We prove a local Holder estimate for any exponent 0 < delta < 1/2 for solutions of the dynamic programming principle u(epsilon) (x) = Sigma(n)(j=1) alpha(j) inf(dim(S)=j) sup(v is an element of s) (vertical bar v vertical bar=1) u(epsilon)(x + epsilon v) + u(epsilon)(x - epsilon v)/2 with alpha(1), alpha(n) > 0 and alpha(2),..., alpha(n-1) >= 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE Sigma(n)(i=1) alpha(i)lambda(i) (D(2)u) = 0, where lambda(1)(D(2)u) <= ... <= lambda(n)(D(2)u) are the eigenvalues of the Hessian.

关键词： dynamic programming principle Holder estimate Viscosity solution Eigenvalue of the Hessian Fully nonlinear PDEs

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Tug-of-war with Kolmogorov

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JOURNAL OF DIFFERENTIAL EQUATIONS 2023年 342卷 501-558页

作者： Fjellstrom, Carmina Nystrom, Kaj Vestberg, Matias Uppsala Univ Dept Math S-75106 Uppsala Sweden

We introduce a new class of strongly degenerate nonlinear parabolic PDEs ((p - 2)Delta(N)(infinity,X) + Delta(X))u(X, Y, t) + (m +p)(X . del(Yu)(X, Y, t) - partial derivative(t)u(X, Y, t)) = 0, (X, Y, t) is an element of R-m x R-m x R, p is an element of (1, infinity), combining the classical PDE of Kolmogorov and the normalized p-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games in-troduced approximate solutions to the stated PDE when the parameter that controls the size of the possible steps goes to zero. Existence and uniqueness of viscosity solutions to the Dirichlet problem is established. The asymptotic mean value property, the associated games and the geometry underlying the Dirichlet prob-lem, all reflect the family of dilation and the Lie group underlying operators of Kolmogorov type, and this makes our setting different from the context of standard parabolic dilations and Euclidean translations applicable in the context of the heat operator and the normalized parabolic infinity Laplace operator. (C) 2022 The Author(s). Published by Elsevier Inc.

关键词： Dirichlet boundary conditions dynamic programming principle p-Laplacian Infinity Laplacian Kolmogorov equation Mean value property Stochastic games Tug-of-war games Viscosity solutions Ultraparabolic Hypoelliptic

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