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VALUE IN MIXED STRATEGIES FOR ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES WITHOUT ISAACS CONDITION

ANNALS OF PROBABILITY 2014年第4期42卷 1724-1768页

作者： Buckdahn, Rainer Li, Juan Quincampoix, Marc Univ Bretagne Occidentale Math Lab CNRS UMR 6205 F-29285 Brest France Shandong Univ Sch Math & Stat Weihai 264209 Shandong Peoples R China

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition pi of the time interval [0, T]. The underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point t(j-i) of the subintervals generated by pi, the controls of Players 1 and 2 are conditionally independent over [t(j-1), t(j)). It is shown that the associated lower and upper value functions W-pi and U-pi converge uniformly on compacts to a function V, the so-called value in mixed strategies, as the mesh of pi tends to zero. This function V is characterized as the unique viscosity solution of the associated Hamilton Jacobi-Bellman-Isaacs equation.

关键词： 2-person zero-sum stochastic differential game Isaacs condition viscosity solution value function backward stochastic differential equations dynamic programming principle randomized controls

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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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Towards Parallelizable Sampling–based Nonlinear Model Predictive Control

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IFAC-PapersOnLine 2017年第1期50卷 13176-13181页

作者： Bobiti, R.V. Lazar, M. Department of Electrical Engineering Eindhoven University of Technology Netherlands

This paper proposes a new sampling–based nonlinear model predictive control (MPC) algorithm, with a bound on complexity quadratic in the prediction horizon N and linear in the number of samples. The idea of the proposed algorithm is to use the sequence of predicted inputs from the previous time step as a warm start, and to iteratively update this sequence by changing its elements one by one, starting from the last predicted input and ending with the first predicted input. This strategy, which resembles the dynamic programming principle, allows for parallelization up to a certain level and yields a suboptimal nonlinear MPC algorithm with guaranteed recursive feasibility, stability and improved cost function at every iteration, which is suitable for real–time implementation. The complexity of the algorithm per each time step in the prediction horizon depends only on the horizon, the number of samples and parallel threads, and it is independent of the measured system state. Comparisons with the fmincon nonlinear optimization solver on a benchmark example indicates that, as the simulation time progresses, the proposed algorithm converges rapidly to the “optimal” solution, even when using a small number of samples. © 2017

关键词： dynamic programming Computational complexity Constrained optimization Cost functions Iterative methods Model predictive control Nonlinear programming Nonlinear systems Predictive control systems Constrained controls dynamic programming principle Embedded model predictive control Nonlinear model predictive control Nonlinear optimization solver Nonlinear predictive control Number of samples Prediction horizon

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ON THE HAMILTON-JACOBI-BELLMAN EQUATIONS

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ACTA APPLICANDAE MATHEMATICAE 1983年第1期1卷 17-41页

作者： LIONS, PL 1. Ceremade Universitè Paris IX-Dauphine Place de Lattre de Tassigny 75775 Paris Cedex 16 France

We consider general problems of optimal stochastic control and the associated Hamilton-Jacobi-Bellman equations. We recall first the usual derivation of the Hamilton-Jacobi-Bellman equations from the dynamic programming principle. We then show and explain various results, including (i) continuity results for the optimal cost function, (ii) characterizations of the optimal cost function as the maximum subsolution, (iii) regularity results, and (iv) uniqueness results. We also develop the recent notion of viscosity solutions of Hamilton-Jacobi-Bellman equations.

关键词： 93E20 35J65 35K60 60J60 Optimal stochastic control diffusion processes Hamilton-Jacobi-Bellman equations viscosity solutions dynamic programming principle

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ASYMPTOTICS OF VALUES IN dynamic GAMES ON LARGE INTERVALS

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ST PETERSBURG MATHEMATICAL JOURNAL 2020年第1期31卷 157-179页

作者： Khlopin, D., V IMM UB RAS NN Krasovskii Inst Math & Mech Ul S Kovalevskoi 16 Ekaterinburg 620999 Russia

dynamic antagonistic games are treated. The dependence of the game value on the payoff function is explored for games with one and the same dynamics, running cost, and possibilities of the players. Each payoff function is viewed as a running cost, averaged in terms of a certain probability distribution on the semiaxis. It is shown that if the dynamic programming principle is fulfilled, then a Tauberian-type theorem holds, specifically, the existence of a uniform limit of the values (as the scaling parameter tends to zero) for games with exponential and/or uniform distribution implies that the same limit exists (as the scaling parameter tends to zero) with arbitrary piecewise-continuous density. Also, a version of this theorem for games with discrete time is proved. The general results are applied to various game settings, both in the deterministic and in the stochastic case. Also, asymptotics are studied as the time horizon tends to infinity.

关键词： dynamic programming principle antagonistic dynamic games differential games Abel mean Cesaro mean

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THE PROBABILISTIC STRUCTURE OF CONTROLLED DIFFUSION-PROCESSES

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ACTA APPLICANDAE MATHEMATICAE 1988年第1期11卷 19-48页

作者： BORKAR, VS TATA INST FUNDAMENTAL RES BANGALORE CTRBANGALORE 560012INDIA MIT INFORMAT & DECIS SYST LABCAMBRIDGEMA 02139

This paper surveys those aspects of controlled diffusion processes wherein the control problem is treated as an optimization problem on a set of probability measures on the path space. This includes: (i) existence results for optimal admissible or Markov controls (both in nondegenerate and degenerate cases), (ii) a probabilistic treatment of the dynamic programming principle, (iii) the corresponding results for control under partial observations, (iv) a probabilistic approach to the ergodic control problem. The paper is expository in nature and aims at giving a unified treatment of several old and new results that evolve around certain central ideas.

关键词： 93E20 49A60 60J60 Stochastic optimal control controlled diffusion processes existence of optimal controls Markov controls dynamic programming principle control under partial observation ergodic control

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Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

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POTENTIAL ANALYSIS 2020年第2期53卷 565-589页

作者： Arroyo, Angel Luiro, Hannes Parviainen, Mikko Ruosteenoja, Eero Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland Norwegian Univ Sci & Technol Dept Math Sci NTNU NO-7491 Trondheim Norway

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in omega subset of Double-struck capital R-n. The method of the proof is based on a game-theoretic i... 详细信息

关键词： dynamic programming principle Local Lipschitz estimates Stochastic games Normalizedp(x)-Laplacian

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Optimal Reinsurance-Investment Strategy Under Risks of Interest Rate, Exchange Rate and Inflation

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2018年第4期20卷 1477-1502页

作者： Guo, Chang Zhuo, Xiaoyang Constantinescu, Corina Pamen, Olivier Menoukeu Nankai Univ Sch Finance Tianjin 300350 Peoples R China Nankai Univ Sch Business Tianjin 300071 Peoples R China Univ Liverpool Inst Financial & Actuarial Math Dept Math Sci Liverpool L69 7ZL Merseyside England African Inst Math Sci Accra Ghana Univ Ghana Accra Ghana

In this paper, we pursue the optimal reinsurance-investment strategy of an insurer who can invest in both domestic and foreign markets. We assume that both the domestic and the foreign nominal interest rates are described by extended Cox-Ingersoll-Ross (CIR) models. In order to hedge the risk associated to investments, rolling bonds, treasury inflation protected securities and futures are purchased by the insurer. We use the dynamic programming principles to explicitly derive both the value function and the optimal reinsurance-investment strategy. As a conclusion, we analyze the impact of the model parameters on both the optimal strategy and the optimal utility.

关键词： Optimal reinsurance-investment strategy Foreign exchange market Extended CIR Stochastic inflation dynamic programming principle 49L20 91G80

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Data-driven surrogates of value functions and applications to feedback control for dynamical systems

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IFAC-PapersOnLine 2018年第2期51卷 307-312页

作者： Schmidt, A. Haasdonk, B. University of Stuttgart Institute for Applied Analysis and Numerical Simulation Pfaffenwaldring 57 Stuttgart70569 Germany

Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since the discretizations typically suffer from the curse of dimensionality. In this paper we introduce a novel approximation technique for the value function of an infinite horizon optimal control. The method is based on solving optimal open loop control problems on a finite horizon with a sampling of the global value function along the generated trajectories. For the interpolation we choose a kernel orthogonal greedy strategy, because these methods are able to produce extreme sparse surrogates and enable rapid evaluations in high dimensions. Two numerical examples prove the performance of the approach and show that the method is able to deal with high-dimensional feedback control problems, where the dimensionality prevents the approximation by most existing methods. © 2018

关键词： dynamic programming dynamical systems Feedback control Numerical methods Approximation techniques Curse of dimensionality dynamic programming principle Feedback control problem Generated trajectories greedy techniques Kernel approximation Optimal feedback control

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Tauberian Theorem for Value Functions

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dynamic GAMES AND APPLICATIONS 2018年第2期8卷 401-422页

作者： Khlopin, Dmitry Russian Acad Sci Krasovskii Inst Math & Mech Ural Branch 16 S Kovalevskaja St Ekaterinburg 620990 Russia Ural Fed Univ Inst Math & Comp Sci Chair Appl Math 4 Turgeneva St Ekaterinburg 620083 Russia

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of -discounted games as the discount tends to zero. We prove that the dynamic programming principle for value functions directly leads to the Tauberian theorem-that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the dynamic programming principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point;a very simple stochastic game model is also considered.

关键词： dynamic programming principle Zero-sum games Abel mean Cesaro mean Differential games

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