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Fuzzy optimal control with application to discounted profit advertising problem

JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2012年第5期23卷 187-192页

作者： Baten, Md. Azizul Kamil, Anton Abdulbasah Univ Utara Malaysia Sch Quantitat Sci Dept Decis Sci Kedah 06010 Malaysia Univ Sains Malaysia Sch Distance Educ Math Program George Town Malaysia

A fuzzy optimal control model was formulated maximizing the expected discounted objective function subject to fuzzy differential equation for fuzzy control system. We proved that the value function of fuzzy optimal control problem satisfied the dynamic programming principle. The optimality equation in fuzzy optimal control was derived and an example was carried out to obtain the optimality conditions by using the optimality equation. As an application, we used the Bellman's principle of optimality to solve the fuzzy advertising model.

关键词： Fuzzy optimal control dynamic programming principle optimality equation profit advertising model

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Solutions of nonlinear PDEs in the sense of averages

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JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 2012年第2期97卷 173-188页

作者： Kawohl, Bernd Manfredi, Juan Parviainen, Mikko Univ Cologne Inst Math D-50923 Cologne Germany Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA Univ Jyvaskyla Dept Math & Stat Jyvaskyla 40014 Finland

We characterize p-harmonic functions including p = 1 and p = infinity by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all p's. We describe a class of random tug-of-war games whose value functions approach p-harmonic functions as the step goes to zero for the full range 1 < p < infinity. (C) 2011 Elsevier Masson SAS. All rights reserved.

关键词： 1-harmonic Asymptotic expansion dynamic programming principle Mean value theorem p-harmonic Stochastic games Two-player zero-sum games Viscosity solutions

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Impulse control problem with switching technology

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STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES 2012年第2-3期84卷 437-460页

作者： Amami, Rim Univ Toulouse 3 Inst Math Toulouse F-31062 Toulouse France

We consider an impulse control problem with switching technology in infinite horizon. We suppose that the firm decides at certain time (impulse time) to switch the technology and the firm value (e.g. a recapitalization). We show that the value function for such problems satisfies a dynamic programming principle. Our objective is to look for an optimal strategy which maximizes the value function.

关键词： impulse control problem maximum conditional profit dynamic programming principle

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Asset allocation under stochastic interest rate with regime switching

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ECONOMIC MODELLING 2012年第4期29卷 1126-1136页

作者： Shen, Yang Siu, Tak Kuen Macquarie Univ Dept Appl Finance & Actuarial Studies Fac Business & Econ Sydney NSW 2109 Australia

We investigate an optimal asset allocation problem in a Markovian regime-switching financial market with stochastic interest rate. The market has three investment opportunities, namely, a bank account, a share and a zero-coupon bond, where stochastic movements of the short rate and the share price are governed by a Markovian regime-switching Vasicek model and a Markovian regime-switching Geometric Brownian motion, respectively. We discuss the optimal asset allocation problem using the dynamic programming approach for stochastic optimal control and derive a regime-switching Hamilton-Jacobi-Bellman (HJB) equation. Particular attention is paid to the exponential utility case. Numerical and sensitivity analysis are provided for this case. The numerical results reveal that regime-switches described by a two-state Markov chain have significant impacts on the optimal investment strategies in the share and the bond. Furthermore, the market prices of risk in both the bond and share markets are crucial factors in determining the optimal investment strategies. (c) 2012 Elsevier B.V. All rights reserved.

关键词： Stochastic interest rate Regime-switching Stochastic flows dynamic programming principle HJB equation

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

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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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Stochastic representation for solutions of Isaacs' type integral-partial differential equations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2011年第12期121卷 2715-2750页

作者： Buckdahn, Rainer Hu, Ying Li, Juan Shandong Univ Weihai Sch Math & Stat Weihai 264200 Peoples R China Univ Bretagne Occidentale Dept Math F-29238 Brest 3 France Univ Rennes 1 IRMAR F-35042 Rennes France Shandong Univ Inst Adv Study Jinan 250100 Peoples R China

In this paper we study the integral partial differential equations of Isaacs' type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral partial differential equations of Hamilton Jacobi Bellman Isaacs' type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs' condition. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Stochastic differential games Poisson random measure Value function Backward stochastic differential equations dynamic programming principle Integral partial differential operators Viscosity solution

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Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths

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POTENTIAL ANALYSIS 2011年第2期34卷 139-161页

作者： Denis, Laurent Hu, Mingshang Peng, Shige Shandong Univ Sch Math Jinan 250100 Peoples R China Univ Evry Val dEssonne Dept Math Equipe Anal & Probabil F-91025 Evry France Fudan Univ Sch Math Shanghai 200433 Peoples R China

In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation-G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.

关键词： Capacity Sublinear expectation G-expectation G-Brownian motion dynamic programming principle

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Applications of Malliavin calculus to the pricing and hedging of Bermudan options

Applications of Malliavin calculus to the pricing and hedgin...

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作者： James Newbury University of Oxford

学位级别：硕士

The pricing of Bermudan options, which give the holder the right to buy or sell an underlying asset at a predetermined price and at a discretely spaced number of times prior to maturity, can be based on a deterministic method or on a probabilistic one. Deterministic methods such as finite differences lose their efficiency as the dimension of the problem increases, and they are there- fore known to suffer from the "curse of dimensionality". Probabilistic methods enable us to overcome this problem by using Monte Carlo simulations. One particular method is the Malliavin pricing and hedging algorithm, which uses representation formulas for conditional expectation and its derivative to approx- imate the price and delta of a Bermudan option. This paper specifically deals with how the powerful tools of Malliavin calculus are applied in the derivation of such representation formulas, and looks at how the latter are subsequently used in the pricing and hedging algorithm.

关键词： Bermudan option dynamic programming principle Malliavin derivative operator Skorohod integral first and second variational processes representation formula localizing function

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AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO TUG-OF-WAR GAMES

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2010年第5期42卷 2058-2081页

作者： Manfredi, Juan J. Parviainen, Mikko Rossi, Julio D. Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA Aalto Univ Sch Sci & Technol FI-00076 Helsinki Finland Univ Alicante Dept Anal Matemat E-03080 Alicante Spain

We characterize solutions to the homogeneous parabolic p-Laplace equation u(t) - vertical bar del u|(2-p)Delta(p)u = (p - 2)Delta(infinity)u + Delta u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these games approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.

关键词： Dirichlet boundary conditions dynamic programming principle parabolic p-Laplacian parabolic mean value property stochastic games tug-of-war games with limited number of rounds viscosity solutions

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UNBOUNDED VISCOSITY SOLUTIONS OF HYBRID CONTROL SYSTEMS

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2010年第1期16卷 176-193页

作者： Barles, Guy Dharmatti, Sheetal Ramaswamy, Mythily Univ Tours Lab Math & Phys Theor F-37200 Tours France Univ Toulouse 3 CNRS UMR 5640 Lab MIP F-31062 Toulouse 9 France TIFR Ctr Applicable Math Bangalore 560065 Karnataka India

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.

关键词： dynamic programming principle viscosity solution quasivariational inequality hybrid control

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