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ON THE ROBUST OPTIMAL STOPPING PROBLEM

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2014年第5期52卷 3135-3175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 study a robust optimal stopping problem with respect to a set P of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its payoff while an adverse player wants to minimize this payoff by choosing an evaluation criteria from P. We show that the upper Snell envelope (Z) over bar of the reward process Y is a supermartingale with respect to an appropriately defined nonlinear expectation E, and (K) over bar is further an (E) under bar- martingale up to the first time tau* when (Z) over bar meets Y. Consequently, tau* is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circumvent using some new methods.

关键词： robust optimal stopping zero-sum game of control and stopping volatility uncertainty dynamic programming principle Snell envelope nonlinear expectation weak stability under pasting path-dependent stochastic differential equations with controls

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A partial history of the early development of continuous-time nonlinear stochastic systems theory

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AUTOMATICA 2014年第2期50卷 303-334页

作者： Kushner, Harold J. Brown Univ Dept Appl Math Providence RI 02912 USA

This article is a survey of the early development of selected areas in nonlinear continuous-time stochastic control. Key developments in optimal control and the dynamic programming principle, existence of optimal controls under complete and partial observations, nonlinear filtering, stochastic stability, the stochastic maximum principle and ergodic control are discussed. Issues concerning wide bandwidth noise for stability, modeling, filtering and ergodic control are dealt with. The focus is on the earlier work, but many important topics are omitted for lack of space. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Stochastic optimal control Viscosity solutions dynamic programming principle Nisio semigroup Existence of optimal controls Existence under partial observations Compactness methods Wide bandwidth noise Approximations to systems Nonlinear filtering Stochastic stability Stochastic maximum principle Ergodic control Stochastic stability Filtering and control with wide bandwidth noise

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The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton-Jacobi-Bellman equations

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SYSTEMS & CONTROL LETTERS 2014年第1期69卷 7-15页

作者： Zhu, Xuehong Nanjing Univ Aeronaut & Astronaut Sch Sci Nanjing 210016 Peoples R China

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Riemannian manifold Hamilton-Jacobi-Bellman equation Backward stochastic differential equations dynamic programming principle Viscosity solution

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Portfolio optimization in a regime-switching market with derivatives

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2014年第1期233卷 184-192页

作者： Fu, Jun Wei, Jiaqin Yang, Hailiang Univ Hong Kong Dept Stat & Actuarial Sci Hong Kong Hong Kong Peoples R China Macquarie Univ Dept Appl Finance & Actuarial Studies Fac Business & Econ Sydney NSW 2109 Australia

We consider the optimal asset allocation problem in a continuous-time regime-switching market. The problem is to maximize the expected utility of the terminal wealth of a portfolio that contains an option, an underlying stock and a risk-free bond. The difficulty that arises in our setting is finding a way to represent the return of the option by the returns of the stock and the risk-free bond in an incomplete regime-switching market. To overcome this difficulty, we introduce a functional operator to generate a sequence of value functions, and then show that the optimal value function is the limit of this sequence. The explicit form of each function in the sequence can be obtained by solving an auxiliary portfolio optimization problem in a single-regime market. And then the original optimal value function can be approximated by taking the limit. Additionally, we can also show that the optimal value function is a solution to a dynamic programming equation, which leads to the explicit forms for the optimal value function and the optimal portfolio process. Furthermore, we demonstrate that, as long as the current state of the Markov chain is given, it is still optimal for an investor in a multiple-regime market to simply allocate his/her wealth in the same way as in a single-regime market. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Functional operator Elasticity approach Portfolio optimization Regime switching dynamic programming principle

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

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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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Shortfall Risk Minimization in Discrete Time Financial Market Models

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SIAM JOURNAL ON FINANCIAL MATHEMATICS 2014年第1期5卷 384-414页

作者： Frikha, N. Univ Paris Diderot LPMA F-75205 Paris France

In this paper, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined on L-p(P) in terms of shortfall risk. Under mild assumptions, namely, the absence of arbitrage opportunity and the nondegeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. In a Markovian framework, the shortfall risk and optimal dynamic strategies are estimated using three main tools: Newton-Raphson Monte Carlo-based procedure, stochastic approximation algorithm, and Markovian quantization scheme. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.

关键词： shortfall risk measures risk minimization dynamic programming principle stochastic approximation algorithm vector quantization Monte Carlo simulation

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On the relationship between the stochastic maximum principle and dynamic programming in singular stochastic control

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STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES 2012年第2-3期84卷 233-249页

作者： Bahlali, Khaled Chighoub, Farid Mezerdi, Brahim Univ Mohamed Khider Lab Appl Math Mexico City 07000 DF Mexico UTV UFR Sci F-83957 La Garde France

This paper investigates the relationship between the stochastic maximum principle and the dynamic programming principle for singular stochastic control problems. The state of the system under consideration is governed by a stochastic differential equation, with nonlinear coefficients, allowing both classical control and singular control. We show that the necessary conditions for optimality, obtained earlier, are in fact sufficient provided some concavity conditions are fulfilled. In a second step, we prove a verification theorem and we show that the solution of the adjoint equation coincides with the derivative of the value function. Finally, using these results, we solve explicitly an example.

关键词： stochastic maximum principle dynamic programming principle singular control

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dynamic programming principle FOR ONE KIND OF STOCHASTIC RECURSIVE OPTIMAL CONTROL PROBLEM AND HAMILTON-JACOBI-BELLMAN EQUATION

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2008年第5期47卷 2616-2641页

作者： Wu, Zhen Yu, Zhiyong Shandong Univ Sch Math & Syst Sci Jinan 250100 Peoples R China Shandong Univ Sch Econ Jinan 250100 Peoples R China

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equation.

关键词： reflected backward stochastic differential equation recursive optimal control problem dynamic programming principle Hamilton-Jacobi-Bellman equation viscosity solution

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LINEARIZATION TECHNIQUES FOR L^∞-CONTROL PROBLEMS AND dynamic programming principleS IN CLASSICAL AND L^∞-CONTROL PROBLEMS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2012年第3期18卷 836-855页

作者： Goreac, Dan Serea, Oana-Silvia Univ Paris Est Marne la Vallee LAMA UMR8050 F-77454 Champs Sur Marne Marne La Vallee France Ecole Polytech CMAP F-91128 Palaiseau France Univ Perpignan LAMPS F-66860 Perpignan France

The aim of the paper is to provide a linearization approach to the L-infinity-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the L-p approach and the associated linear formulations. This seems to be the most appropriate tool for treating L-infinity problems in continuous and lower semicontinuous setting.

关键词： dynamic programming principle essential supremum HJ equations occupational measures L-p approximations

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NASH EQUILIBRIUM PAYOFFS FOR STOCHASTIC DIFFERENTIAL GAMES WITH REFLECTION

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2013年第4期19卷 1189-1208页

作者： Lin, Qian Univ Bielefeld Ctr Math Econ D-33501 Bielefeld Germany

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.

关键词： Backward stochastic differential equations dynamic programming principle Nash equilibrium payoffs stochastic differential games

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