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STOCHASTIC OPTIMAL CONTROL PROBLEM WITH INFINITE HORIZON DRIVEN BY G-BROWNIAN MOTION

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2018年第2期24卷 873-899页

作者： Hu, Mingshang Wang, Falei Shandong Univ Zhongtai Secur Inst Financial Studies Jinan 250100 Shandong Peoples R China Shandong Univ Inst Adv Res Jinan 250100 Shandong Peoples R China

The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation.

关键词： G-Brownian motion backward stochastic differential equations stochastic optimal control dynamic programming principle

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OPTIMAL INVENTORY CONTROL WITH JUMP DIFFUSION AND NONLINEAR dynamicS IN THE DEMAND

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2018年第1期56卷 53-74页

作者： Liu, Jingzhen Yiu, Ka Fai Cedric Bensoussan, Alain Cent Univ Finance & Econ Sch Insurance Beijing 100081 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China City Univ Hong Kong Dept Syst Engn & Engn Management Kowloon Hong Kong Peoples R China Univ Texas Dallas Jindal Sch Management Dallas TX 75083 USA

In this paper, we consider an inventory control problem with a nonlinear evolution and a jump-diffusion demand model. This work extends the earlier inventory model proposed by Benkherouf and Johnson [Math. Methods Oper. Res., 76 (2012), pp. 377-393] by including a general jump process. However, as those authors note, their techniques are not applicable to models with demand driven by jump-diffusion processes with drift. Therefore, the combination of diffusion and general compound Poisson demands is not completely solved. From the dynamic programming principle, we transform the problem into a set of quasi-variational inequalities (Q.V.I.). The difficulty arises when solving the Q.V.I. because the second derivative and integration term appear in the same inequality. Our technique is to construct a set of coupled auxiliary functions. Then, an analytical study of the Q.V.I. implies the existence and uniqueness of an (s, S) policy.

关键词： inventory control jump diffusion dynamic programming principle quasi-variational inequalities

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Stochastic differential games with competing Brownian particles and related Isaacs' equations

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OPTIMAL CONTROL APPLICATIONS & METHODS 2018年第2期39卷 519-536页

作者： Feng, Xinwei Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

In this paper, we study zero-sum two-player stochastic differential games in which the state equations are competing Brownian particles and the cost functional is defined by generalized backward stochastic differential equations with more than one increasing process. After we study the regularity of competing Brownian particles, we establish the dynamic programming principle for the upper and lower value functions and show that these are the unique viscosity solution of the associated upper and lower Isaacs' equations, which are fully nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions.

关键词： competing Brownian particles dynamic programming principle generalized backward stochastic differential equations partial differential equations stochastic differential games

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Singular Linear Quadratic Optimal Control Problem for Stochastic Nonregular Descriptor Systems

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ASIAN JOURNAL OF CONTROL 2018年第5期20卷 1782-1792页

作者： Wang, Xin Liu, Bin Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Hubei Peoples R China

This paper is concerned with the singular linear quadratic (SLQ) optimal control problem for stochastic nonregular descriptor systems with time-delay. By means of some reasonable assumptions and a series of equivalent transformations, the problem is finally transformed into a positive linear quadratic (LQ) problem for standard stochastic systems. Then dynamic programming principle is used to establish the solvability of the original problem, and the desired explicit presentation of the optimal controller is given in terms of matrix iterative form. The results due to Feng etal. are generalized and improved. As an application, a numerical example is presented to demonstrate the efficiency of the proposed approach.

关键词： LQ optimal control nonregular descriptor systems stochastic systems dynamic programming principle

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Optimal bounded parametric control for random vibration of dielectric elastomer balloon

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NONLINEAR dynamicS 2018年第2期94卷 1081-1093页

作者： Jin, Xiaoling Tian, Yanping Wang, Yong Huang, Zhilong Zhejiang Univ Key Lab Soft Machines & Smart Devices Zhejiang Pr Dept Engn Mech Hangzhou Zhejiang Peoples R China Hangzhou Dianzi Univ Coll Mech Engn Hangzhou Zhejiang Peoples R China

As a functional device, dielectric elastomer balloon with surface electrodes operates under impressed pressure and voltage. The unavoidable pressure disturbance will induce prominent vibration around the equilibrium position, which may deteriorate its operating performance and accelerate its failure. This manuscript concentrates on the vibration suppression by slightly adjusting the voltage under certain given constraint. The displacement perturbation is governed by a nonlinear stochastic differential equation which is determined by expanding technique at equilibrium points. The pressure disturbance described by ideal Gaussian white noise is a parametric excitation and the function of control voltage directly multiplies with a function of the displacement perturbation. This comes down to an optimal bounded parametric control problem. The optimal control is first formally determined by the extremum condition in dynamic programming equation. The nonlinear dynamic programming equation is then approximately solved by the pseudo-inverse algorithm. The good control effectiveness and high robustness to the intensity of pressure disturbance are verified by numerical examples. Particularly, the optimal control strategy with not too small bound can effectively avoid the interwell motion for the cases with bistable potential.

关键词： Dielectric elastomer balloon Random vibration Optimal bounded control Parametric control dynamic programming principle Coulomb friction

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Stochastic Maximum principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

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Chinese Annals of Mathematics,Series B 2018年第5期39卷 773-790页

作者： Siyu LV Zhen WU School of Mathematics Southeast UniversityNanjing 211189China School of Mathematics Shandong UniversityJinan 250100China

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.

关键词： Stochastic maximum principle dynamic programming principle Forward-backward stochastic differential equation Regime switching Jump diffusion

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Regularity properties in a state-constrained expected utility maximization problem

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2018年第2期88卷 185-240页

作者： Lazgham, Mourad Univ Mannheim Dept Math Mannheim Germany

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.

关键词： Expected utility maximization problem Value function Price impact Optimal strategy dynamic programming principle Bellman's principle Hamilton-Jacobi-Bellman equation Viscosity solution Comparison principle

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Regularity for nonlinear stochastic games

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ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 2018年第6期35卷 1435-1456页

作者： Luiro, Hannes Parviainen, Mikko Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland

We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations. (C) 2018 Elsevier Masson SAS. All rights reserved.

关键词： dynamic programming principle p-Laplace Tug-of-war Tug-of-war with noise with space dependent probabilities Viscosit solutions

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Approximation of optimal ergodic dividend strategies using controlled Markov chains

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IET CONTROL THEORY AND APPLICATIONS 2018年第16期12卷 2194-2204页

作者： Jin, Zhuo Yang, Hailiang Yin, George Univ Melbourne Dept Econ Ctr Actuarial Studies Melbourne Vic 3010 Australia Univ Hong Kong Dept Stat & Actuarial Sci Hong Kong Hong Kong Peoples R China Wayne State Univ Dept Math Detroit MI 48202 USA

This study develops a numerical method to find optimal ergodic (long-run average) dividend strategies in a regime-switching model. The surplus process is modelled by a regime-switching process subject to liability constraints. The regime-switching process is modelled by a finite-time continuous-time Markov chain. Using the dynamic programming principle, the optimal long-term average dividend payment is a solution to the coupled system of Hamilton-Jacobi-Bellman equations. Under suitable conditions, the optimal value of the long-term average dividend payment can be determined by using an invariant measure. However, due to the regime switching, getting the invariant measure is very difficult. The objective is to design a numerical algorithm to approximate the optimal ergodic dividend payment strategy. By using the Markov chain approximation techniques, the authors construct a discrete-time controlled Markov chain for the approximation, and prove the convergence of the approximating sequences. A numerical example is presented to demonstrate the applicability of the algorithm.

关键词： continuous time systems approximation theory Markov processes dynamic programming discrete time systems optimal control discrete-time controlled Markov chain optimal ergodic dividend strategies controlled Markov chains numerical method regime-switching model surplus process regime-switching process subject finite-time continuous-time Markov chain dynamic programming principle long-term average dividend payment Hamilton-Jacobi-Bellman equations optimal value invariant measure regime switching optimal ergodic dividend payment strategy Markov chain approximation techniques

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Robust stochastic control modeling of dam discharge to suppress overgrowth of downstream harmful algae

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APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY 2018年第3期34卷 338-354页

作者： Yoshioka, Hidekazu Yaegashi, Yuta Shimane Univ Fac Life & Environm Sci Matsue Shimane 6908504 Japan Kyoto Univ Grad Sch Agr Kyoto Japan Japan Soc Promot Sci Kyoto Japan

The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal of the problem is to find a fit-for-purpose and environmentally sound operation policy of the flow discharge from a dam so that overgrowth of the harmful algae Cladophora glomerataKutzing in its downstream river is effectively suppressed. A minimal stochastic differential equation for the algae growth dynamics with uncertain growth rate is first presented. The performance index to be maximized by the operator of the dam while minimized by nature is formulated within the framework of differential games. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs equation whose solution determines the worst-case optimal operation policy of the dam, ie, the policy that the operator wants to find. Application of the model to overgrowth suppression of Cladophora glomerataKutzing just downstream of a dam in a Japanese river is then carried out. Values of the model parameters are identified with which the model successfully reproduces the observed population dynamics. A series of numerical experiments are performed to find the most effective operation policy of the dam based on a relaxation of the current policy.

关键词： Cladophora glomerata Kutzing dynamic programming principle flow discharge Hamilton-Jacobi-Bellman-Isaacs equation model misspecification multiplier robust control

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