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Relationship Between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions

APPLIED MATHEMATICS AND OPTIMIZATION 2011年第2期63卷 151-189页

作者： Shi, Jing-Tao Wu, Zhen Shandong Univ Sch Math Jinan 250100 Peoples R China

This paper is concerned with the stochastic optimal control problem of jump diffusions. The relationship between stochastic maximum principle and dynamic programming principle is discussed. Without involving any derivatives of the value function, relations among the adjoint processes, the generalized Hamiltonian and the value function are investigated by employing the notions of semijets evoked in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is optimal.

关键词： Jump diffusions Stochastic optimal control Maximum principle dynamic programming principle Verification theorem Viscosity solution

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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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SOBOLEV WEAK SOLUTIONS OF THE HAMILTON-JACOBI-BELLMAN EQUATIONS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2014年第3期52卷 1499-1526页

作者： Wei, Lifeng Wu, Zhen Zhao, Huaizhong Ocean Univ China Sch Math Sci Qingdao 266003 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China Univ Loughborough Dept Math Sci Loughborough LE11 3TU Leics England

This paper is concerned with the Sobolev weak solutions of the Hamilton-Jacobi-Bellman (HJB) equations. These equations are derived from the dynamic programming principle in the study of stochastic optimal control problems. Adopting the Doob-Meyer decomposition theorem as one of the main tools, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. In the recursive optimal control problem, the cost function is described by the solution of a backward stochastic differential equation (BSDE). This problem has a practical background in economics and finance. We prove that the value function is the unique Sobolev weak solution of the related HJB equation by virtue of the nonlinear Doob-Meyer decomposition theorem introduced in the study of BSDEs.

关键词： dynamic programming principle Hamilton-Jacobi-Bellman equations Sobolev weak solutions backward stochastic differential equations Doob-Meyer decomposition theorem

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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 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 stopping with random maturity under nonlinear expectations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2017年第8期127卷 2586-2629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 analyze an optimal stopping problem sup(gamma is an element of T) (xi) over bar 0[y(gamma Lambda tau 0)] with random maturity to under a nonlinear expectation (xi) over bar0[.] := sup(P is an element of P) Eg[.], where P is a weakly compact set of mutually singular probabilities. The maturity tau(0) is specified as the hitting time to level 0 of some continuous index process X at which the payoff process g is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow. We utilize a martingale approach to construct an optimal pair (P-*, y(*)) for sup((P, gamma)is an element of P X T) Ep[y(gamma Lambda tau 0)], in which y(*) is the first time y meets the limit. L of its approximating (xi) over bar -Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process y, we approximate tau(0) by an increasing sequence of Lipschitz continuous stopping times and approximate y by a sequence of uniformly continuous processes. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Discretionary stopping Random maturity Controls in weak formulation Optimal stopping Nonlinear expectation Weak stability under pasting Lipschitz continuous stopping time dynamic programming principle Martingale approach

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A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

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APPLIED MATHEMATICS AND OPTIMIZATION 2014年第2期70卷 253-278页

作者： Hu, Mingshang Ji, Shaolin Yang, Shuzhen Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China Shandong Univ Qilu Inst Finance Jinan 250100 Shandong Peoples R China

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

关键词： G-expectation Backward stochastic differential equations Stochastic optimal control dynamic programming principle Viscosity solution

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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 differential games with controlled regime-switching

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第4期43卷 1-27页

作者： Ma, Chenglin Zhao, Huaizhong Shandong Univ Sch Math Jinan 250100 Peoples R China Univ Durham Dept Math Sci Durham DH1 3LE England Shandong Univ Res Ctr Math & Interdisciplinary Sci Qingdao 266237 Peoples R China

In this article, we consider a two-player zero-sum stochastic differential game with regime-switching. Different from the results in existing literature on stochastic differential games with regime-switching, we consider a game between a Markov chain and a state process which are two fully coupled stochastic processes. The payoff function is given by an integral with random terminal horizon. We first study the continuity of the lower and upper value functions under some additional conditions, based on which we establish the dynamic programming principle. We further prove that the lower and upper value functions are unique viscosity solutions of the associated lower and upper Hamilton-Jacobi-Bellman-Isaacs equations with regime-switching, respectively. These two value functions coincide under the Isaacs condition, which implies that the game admits a value. We finally apply our results to an example.

关键词： Stochastic differential games Controlled regime-switching dynamic programming principle Viscosity solutions Hamilton-Jacobi-Bellman-Isaacs equations

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ON THE ROBUST OPTIMAL STOPPING PROBLEM

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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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