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dynamic APPROACHES FOR SOME TIME-INCONSISTENT OPTIMIZATION PROBLEMS

ANNALS OF APPLIED PROBABILITY 2017年第6期27卷 3435-3477页

作者： Karnam, Chandrasekhar Ma, Jin Zhang, Jianfeng Univ Southern Calif Dept Math Los Angeles CA 90089 USA

In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time consistency as well as any method that is based on Pontryagin's maximum principle. The fundamental obstacle is the dilemma of having to invoke the dynamic programming principle (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the "dynamic utility" under which the original time-inconsistent problem (under the fixed utility) becomes a time-consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases;and we argue that the time inconsistency is essentially equivalent to the lack of comparison principle. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.

关键词： Time inconsistency dynamic programming principle stochastic maximum principle comparison principle duality dynamic utility master equation path derivative

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Stochastic modeling and control of bioreactors

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IFAC-PapersOnLine 2017年第1期50卷 12611-12616页

作者： Fontbona, J. Ramírez C., H. Riquelme, V. Silva, F.J. Universidad de Chile Beauchef 851 Casilla 170-3 Santiago 3 Chile Institut de recherche XLIM-DMI UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges Limoges87060 France

In this work we propose a stochastic model for a sequencing-batch reactor (SBR) and for a chemostat. Both models are described by systems of Stochastic Differential Equations (SDEs), which are obtained as limits of suitable Markov Processes characterizing the microscopic behavior. We study the existence of solutions of the obtained equations as well as some properties, among which the possible extinction of the biomass is the most remarkable feature. The implications of this behavior are illustrated in the problem consisting in maximizing the probability of reaching a desired depollution level prior to biomass extinction. © 2017

关键词： Stochastic models Batch reactors Chemostats Differential equations dynamic programming Markov processes Pollution control Stochastic control systems Stochastic systems Demographic stochasticity Depollution dynamic programming principle Existence of Solutions Microscopic behavior Sequencing batch reactors Stochastic control Systems of stochastic differential equations

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MAXIMAL OPERATORS FOR THE p-LAPLACIAN FAMILY

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PACIFIC JOURNAL OF MATHEMATICS 2017年第2期287卷 257-295页

作者： Blanc, Pablo Pinasco, Juan P. Rossi, Julio D. Univ Buenos Aires FCEyN Dept Matemat Ciudad UnivPabellon 1 RA-1428 Buenos Aires DF Argentina

We prove existence and uniqueness of viscosity solutions for the problem max {-Delta(p1)u(x), -Delta(p2)u(x)} = f(x) in a bounded smooth domain Omega subset of R-N with u = g on partial derivative Omega. Here -Delta(p)u = (N + p)(-1) vertical bar Du vertical bar(2-p) div (vertical bar Du vertical bar(p-2)Du) is the 1-homogeneous p-Laplacian and we assume that 2 <= p(1), p(2) <= infinity. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-of-war game (without noise) or playing at random. Moreover, the operator max {-Delta(p1)u(x), -Delta(p2)u(x)} provides a natural analogue with respect to p-Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.

关键词： Dirichlet boundary conditions dynamic programming principle p-Laplacian tug-of-war games

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Two Different Approaches for Optimal Control Problem of Fully Coupled FBSDEs

Two Different Approaches for Optimal Control Problem of Full...

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第37届中国控制会议

作者： Jingtao Shi School of Mathematics Shandong University

This paper is concerned with the optimal control problem,where the recursive cost functional is defined as one of the solution to a controlled fully coupled forward-backward stochastic differential equation（FBSDE）,and the control domain is *** different approaches–dynamic programming principle（DPP） and maximum principle（MP）–are applied to solve the problem and the relationship between them are *** some differentiable assumptions,relations among the adjoint processes,the value function and the generalized Hamiltonian function are proved,whereas the diffusion term of the forward equation is independent of the state variable *** general case for the problem is open.A linear example is discussed as the illustration of our main result.

关键词： Stochastic optimal control fully coupled FBSDE dynamic programming principle maximum principle

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CONNECTION BETWEEN MP AND DPP FOR STOCHASTIC RECURSIVE OPTIMAL CONTROL PROBLEMS: VISCOSITY SOLUTION FRAMEWORK IN THE GENERAL CASE

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第5期55卷 3258-3294页

作者： Nie, Tianyang Shi, Jingtao Wu, Zhen Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

This paper deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex. We focus on the connection between the general maximum principle and the dynamic programming principle for such a control problem without the assumption that the value is smooth enough;the set inclusions among the sub-and super-jets of the value function and the first-order and second-order adjoint processes as well as the generalized Hamiltonian function are established. Moreover, by comparing these results with the classical ones in Yong and Zhou [Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999], it is natural to obtain the first- and second-order adjoint equations of Hu [Probability, Uncertainty and Quantitative Risk, 2 (2017), 1].

关键词： stochastic recursive optimal control backward stochastic differential equation maximum principle dynamic programming principle subjets superjets viscosity solution

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Stochastic modeling and control of bioreactors

Stochastic modeling and control of bioreactors

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20th World Congress of the International-Federation-of-Automatic-Control (IFAC)

作者： Fontbona, J. Ramirez, H. C. Riquelme, V. Silva, F. J. Univ Chile Dept Ingn Matemat Beauchef 851Casilla 170-3 Santiago 3 Chile Univ Chile CNRS Ctr Modelamiento Matemat UMI 2807 Beauchef 851Casilla 170-3 Santiago 3 Chile Univ Limoges Fac Sci & Tech UMR CNRS 7252 Inst Rech XLIM DMI F-87060 Limoges France

关键词： Chemostat SBR demographic stochasticity Stochastic Control dynamic programming principle

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Capital inflow-terms of trade 'nexus': Does it lead to financial crisis?

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ECONOMIC MODELLING 2017年 65卷 18-29页

作者： Basak, Gopal K. Das, Pranab Kumar Rohit, Allena Indian Stat Inst Stat Math Unit Kolkata 700108 India Ctr Studies Social Sci R1 Kolkata 700094 India Goizueta Business Sch Atlanta GA 30322 USA

The paper models the nexus of foreign capital inflow and dynamic terms of trade to explain financial crisis in the form of sudden stop or reversal of capital inflow. Crisis in this structure is rooted in the role played by dynamic terms of trade rather than informational imperfections as generally found in the existing literature. Inspite of satisfying the regularity conditions for model consistency episodes of sudden crises get magnified due to the non-linearity of the equilibrium relations. This is the novelty of this paper and differentiates it from the standard theoretical literature, and well captures empirical evidence documented in the literature. Non-linearity plays a very important role in the model. Expectation of the exchange rate depreciation has higher potential to generate a financial crisis than shift in the risk perception of foreign lenders or supply shock in the borrowing country.

关键词： Capital inflow dynamic terms of trade Financial crisis Interest rate determination dynamic programming principle Portfolio theory E-M algorithm

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A PSEUDO-MARKOV PROPERTY FOR CONTROLLED DIFFUSION PROCESSES

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2016年第2期54卷 1017-1029页

作者： Claisse, Julien Talay, Denis Tan, Xiaolu INRIA Sophia Antipolis 2004 Route LuciolesBP93 Sophia Antipolis France Univ Paris 09 CEREMADE Pl Marechal De Lattre De Tassigny F-75775 Paris 16 France

In this note, we propose two different approaches to rigorously justify a pseudo-Markov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314]. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.

关键词： stochastic control martingale problem dynamic programming principle pseudo-Markov property

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Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one-dimensional rivers

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MATHEMATICAL BIOSCIENCES 2016年第0期281卷 139-148页

作者： Yoshioka, Hidekazu Shimane Univ Fac Life & Environm Sci Nishikawatsu Cho 1060 Matsue Shimane 6908504 Japan

Upstream migration of fish schools in 1-D rivers as an optimal control problem is formulated where their swimming velocity and the horizontal oblateness are taken as control variables. The objective function to be maximized through a migration process consists of the biological and ecological profit to be gained at the upstream-end of a river, energetic cost of swimming against the flow, and conceptual cost of forming a school. Under simplified conditions where the flow is uniform in both space and time and the profit to be gained at the goal of migration is sufficiently large, the optimal control variables are determined from a system of algebraic equations that can be solved in a cascading manner. Mathematical analysis of the system reveals that the optimal controls are uniquely found and the model is exactly solvable under certain conditions on the functions and parameters, which turn out to be realistic and actually satisfied in experimental fish migration. Identification results of the functional shapes of the functions and the parameters with experimentally observed data of swimming schools of Plecoglossus altivelis (Ayu) validate the present mathematical model from both qualitative and quantitative viewpoints. The present model thus turns out to be consistent with the reality, showing its potential applicability to assessing fish migration in applications. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Fish school Upstream migration Optimal control problem dynamic programming principle Hamilton-jacobi-Bellman equation

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FULLY COUPLED FORWARD-BACKWARD SDES INVOLVING THE VALUE FUNCTION AND ASSOCIATED NONLOCAL HAMILTON-JACOBI-BELLMAN EQUATIONS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2016年第2期22卷 519-538页

作者： Hao, Tao Li, Juan Shandong Univ Sch Math & Stat Weihai 264209 Peoples R China Shandong Univ Tradit Chinese Med Sch Sci & Technol Jinan 250355 Peoples R China

A new type of controlled fully coupled forward-backward stochastic differential equations is discussed, namely those involving the value function. With a new iteration method, we prove an existence and uniqueness theorem of a solution for this type of equations. Using the notion of extended "backward semigroup", we prove that the value function satisfies the dynamic programming principle and is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman equation.

关键词： Fully coupled FBSDE involving value function dynamic programming principle fully coupled mean-field FBSDE viscosity solution nonlocal Hamilton-Jacobi-Bellman equation

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