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BACKWARD STOCHASTIC RICCATI EQUATION WITH JUMPS ASSOCIATED WITH STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL WITH JUMPS AND RANDOM COEFFICIENTS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2020年第1期58卷 393-424页

作者： Zhang, Fu Dong, Yuchao Meng, Qingxin Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Univ Angers Dept Math 2 Bd Lavoisier F-49045 Angers 01 France Fudan Univ Dept Math Shanghai 200433 Peoples R China Huzhou Univ Dept Math Huzhou 313000 Zhejiang Peoples R China

In this paper, we investigate the solvability of matrix valued backward stochastic Riccati equations with jumps (BSREJ), which are associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and a Poisson process. By the dynamic programming principle, Doob-Meyer decomposition, and inverse flow technique, the existence and uniqueness of the solution for the BSREJ are established. The difficulties addressed in this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that (i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson process may not exist without additional technical conditions, and (ii) showing that the inverse matrix term involving a jump process in the generator is well-defined. Utilizing the structure of the optimal control problem, we overcome these difficulties and establish the existence of the solution. In addition, we show the construction of the optimal feedback control with the help of the Riccati equation and the relation between the solution of the Riccati equation and the value function of the SLQ problem, which also implies the uniqueness of BSREJ.

关键词： dynamic programming principle Doob-Meyer decomposition stochastic differential equation Poisson process backward stochastic Riccati equation with jumps

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Coupled dynamics with an external system and application to international finance

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2019年 520卷 409-432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 BP Township Kolkata 700094 India Emory Univ Goizueta Business Sch Atlanta GA 30322 USA

The paper develops a generalized model of coupled dynamics for addressing the choice theoretic problems of economics and other behavioural sciences. The model extends the framework of coupled system to include an external sector that generates a richer dynamics. The model is applied to explain foreign capital inflow from the developed to the developing countries. Under certain regularity conditions, the existence of the solutions to the dynamic choice problem is proved and they are then obtained by numerical technique because of the non-linearity of the related functions. Robustness results are achieved via additional simulations by perturbation of the baseline parameter values. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Coupled dynamics Capital inflow dynamic programming principle Portfolio theory E-M algorithm dynamic terms of trade

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Second order Hamilton-Jacobi-Bellman equations with an unbounded operator

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第13期75卷 4784-4797页

作者： Zalinescu, Adrian Acad Romana O Mayer Inst Math Iasi 700505 Romania

This work is devoted to the study of a class of Hamilton-Jacobi-Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Hamilton-Jacobi-Bellman equations dynamic programming principle Optimal stochastic control Multivalued stochastic differential equations

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Random vibration control for multi-degree-of-freedom mechanical systems with soft actuators

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2019年 113卷 44-54页

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

Soft robotics include soft actuators and possess the capacity of large deformation and environmental compatibility. Weak environmental disturbances may deteriorate the operating performance of soft robotics due to the low stiffness of soft actuators. This manuscript investigates multi-degree-of-freedom nonlinear mechanical systems with dielectric elastomer actuators and establishes the bounded/unbounded optimal control strategies to suppress the random vibration around the equilibrium position by adjusting the imposed voltage in real time. First, the constitutive relation of a plan-type dielectric elastomer actuator and then the vibrating equation of the multi-degree-of-freedom nonlinear system around the equilibrium position are derived successively. The bounded/unbounded optimal control problems are then established by adopting the corresponding performance indexes. The bounded/unbounded optimal control strategies are then derived by combining the stochastic averaging technique and stochastic dynamic programming principle. Numerical results on a two-degree-of-freedom nonlinear system illustrate the application and efficacy of the proposed optimal control strategies.

关键词： Random vibration Optimal control Mechanical system Soft actuator Stochastic averaging dynamic programming principle

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

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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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The corporate optimal portfolio and consumption choice problem in the real project with borrowing rate higher than deposit rate

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期175卷 1596-1608页

作者： Wu, Zhen Zhang, Liyan Shandong Univ Sch Math & Syst Sci Jinan 250100 Peoples R China

In this paper, one kind of corporate optimal portfolio and consumption choice problem is studied for a investor who can invest his wealth in the bond (bank account) and in a real project which has the production. The bank pays at an interest rate for any deposit and takes at a large rate for any loan. The optimal strategies are obtained by Hamilton-Jacobi-Bellman equation which is derived from dynamic programming principle. We also give the economic analysis to the optimal choice using the investment theory. For the specific Hyperbolic Absolute Risk Aversion case, we get the explicit optimal investment and consumption solution. At last, we give some simulation results to illustrate the optimal result and the influence of the volatility parameter on the optimal choice. (c) 2005 Elsevier Inc. All rights reserved.

关键词： consumption/investment optimization dynamic programming principle HJB equations stochastic control hyperbolic absolute risk aversion

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Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2009年第4期70卷 1776-1796页

作者： Li, Juan Peng, Shige Shandong Univ Dept Math Weihai 264200 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Shandong Univ Sch Math & Syst Sci Jinan 250100 Peoples R China

In this paper we Study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L, Wu. Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for Value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton-Jacobi-Bellman equations with integral-differential operators. For this proof, we adapt Peng's BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Poisson random measure Value function Backward stochastic differential equations dynamic programming principle Viscosity solution Stochastic backward semigroup

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Transition density function expansion methods for portfolio optimization

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OPTIMAL CONTROL APPLICATIONS & METHODS 2024年第4期45卷 1345-1374页

作者： Lu, Yuxuan Zhou, Qing Wu, Weixing Xiao, Weilin Beijing Univ Posts & Telecommun Sch Sci Minist Educ Key Lab Math & Informat Networks Beijing 100876 Peoples R China Capital Univ Econ & Business Sch Finance Beijing Peoples R China Zhejiang Univ Sch Management Hangzhou Peoples R China

In this study, we introduce transition density function expansion methods inspired from Yang et al. (J Econom. 2019;209(2):256-288.) to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Utilizing Bellman's dynamic programming principle, we initially recast the conditional expectation via the transition density function pertinent to the diffusion process. Subsequently, we employ the Ito-Taylor expansion and Delta expansion techniques to the transition density function associated with the multivariate diffusion process, facilitated by a quasi-Lamperti transformation, aiming to derive explicit recursive expressions for expansion coefficient functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies, achieved through discretization methodologies with rigorous proof of expansion convergence in portfolio optimization. Both theoretical and practical demonstrations are presented to validate the convergence of these approximate techniques in addressing stochastic control challenges. To underscore the efficiency and precision of our proposed methods, we apply them to portfolio selection problems within several benchmark models, and highlight the reduced complexity in comparison to the current methodologies. We introduce transition density function expansion methods to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies with proof of expansion convergence in portfolio optimization. We apply them to portfolio selection problems within several benchmark models, highlighting the reduced complexity in comparison to the

关键词： dynamic programming principle portfolio optimization quasi-Lamperti transformation transition density expansion

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Stochastic recursive optimal control problem with mixed delay under viscosity solution's framework

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OPTIMAL CONTROL APPLICATIONS & METHODS 2021年第2期42卷 445-468页

作者： Meng, Weijun Shi, Jingtao Shandong Univ Sch Math Jinan 250100 Peoples R China

This article is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super- and sub-jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.

关键词： dynamic programming principle maximum principle mixed delay stochastic recursive optimal control verification theorem viscosity solution

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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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