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 Applied Mathematical Sciences , 2013, Abstract: This paper considers a discrete dynamic model of profit optimizationwith risk-free and risk assets. Optimal control strategy is definedby quadratic functional, minimization of which aims at three objectives:tracking of reference portfolio, minimization of investment at each stepand maximization of the final income. The mathematical model is alinear–quadratic discrete optimal control problem with control restrictions.By using the approaches proposed in the paper [1] and [2], we deducenecessary Pontryagin-type optimality conditions to obtain a closedform solution for the optimal control via conjugate variables. It enablesus to reduce the boundary value problem to a finite system of linearalgebraic equations.
 系统工程理论与实践 , 2007, Abstract: Considering the constrain on VaR,laws,regulations and operation,using portfolio profits maximum of bank as objective function,applying the Backward Induction idea of multi-period assets allocation and Linear Programming Method,the dynamic optimization model of bank asset portfolio is set up.There are some characteristics and innovations in this paper.The characteristics lie on four aspects. Firstly,by using Backward Induction Method,the portfolio of this period is set up based on the portfolio of the next period.And then such problems as the neglect or lack consideration of the interaction of each period can be solved.Secondly,it is taken into account that the earning rate of this period will be affected by loan credit migration of the former period.Then in this study the earning rates of different levels and the probability of risk migration for one year are used to obtain the earning rates of every corporation each year and mean square deviations.It can objectively reflect the actual portfolio and risk.As a result the neglect of the earning rates is avoided.Finally,the portfolio risk of multi-period loan is controlled by the introduction of VaR constrain.Then the lack consideration of the bank's risk tolerance ability and the demand of capital intendance in present multi-period study are avoided.
 Quantitative Finance , 2014, Abstract: In this work, we consider the optimal portfolio selection problem under hard constraints on trading volume amounts when the dynamics of the risky asset returns are governed by a discrete-time approximation of the Markov-modulated geometric Brownian motion. The states of Markov chain are interpreted as the states of an economy. The problem is stated as a dynamic tracking problem of a reference portfolio with desired return. We propose to use the model predictive control (MPC) methodology in order to obtain feedback trading strategies. Our approach is tested on a set of a real data from the radically different financial markets: the Russian Stock Exchange MICEX, the New York Stock Exchange and the Foreign Exchange Market (FOREX).
 Cristinca Fulga Lecture Notes in Engineering and Computer Science , 2009, Abstract:
 Erik Taflin Mathematics , 1999, Abstract: A quadratic discrete time probabilistic model, for optimal portfolio selection in (re-)insurance is studied. For positive values of underwriting levels, the expected value of the accumulated result is optimized, under constraints on its variance and on annual ROE's. Existence of a unique solution is proved and a Lagrangian formalism is given. An effective method for solving the Euler-Lagrange equations is developed. The approximate determination of the multipliers is discussed. This basic model is an important building block for more complete models.
 Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/802518 Abstract: We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient. 1. Introduction In an insurance business, a reinsurance arrangement is an agreement between an insurer and a reinsurer under which claims are split between them in an agreed manner. Thus, the insurer (cedent company) is insuring part of a risk with a reinsurer and pays premium to the reinsurer for this cover. Reinsurance can reduce the probability of suffering losses and diminish the impact of the large claims of the company. Proportional reinsurance is one of the reinsurance arrangement, which means the insurer pays a proportion, say , when the claim occurs and the remaining proportion, , is paid by the reinsurer. If the proportion can be changed according to the risk position of the insurance company, this is the dynamic proportional reinsurance. Researches dealing with this problem in the one-dimensional risk model have been done by many authors. See for instance, H？jgaard and Taksar [1, 2], Schmidli [3] considered the optimal proportional reinsurance policies for diffusion risk model and for compound Poisson risk model, respectively. Works combining proportional and other type of reinsurance polices for the diffusion model were presented in Zhang et al. [4]. If investment or dividend can be involved, this problem was discussed by Schmidli [5] and Azcue and Muler [6], respectively. References about dynamic reinsurance of large claim are Taksar and Markussen [7], Schmidli [8], and the references therein. Although literatures on the optimal control are increasing rapidly, seemly that none of them consider this problem in the multidimensional risk model so far. This kind of model depicts that an unexpected claim event usually triggers several types of claims in an umbrella insurance policy, which means that a single event influences the risks of the entire portfolio. Such risk model has become more important for the insurance companies due to the fact that it is useful when the insurance companies
 Journal of Probability and Statistics , 2009, DOI: 10.1155/2009/451856 Abstract: The paper revisits the classical problem of premium rating within a heterogeneous portfolio of insurance risks using a continuous stochastic control framework. The portfolio is divided into several classes where each class interacts with the others. The risks are modelled dynamically by the means of a Brownian motion. This dynamic approach is also transferred to the design of the premium process. The premium is not constant but equals the drift of the Brownian motion plus a controlled percentage of the respective volatility. The optimal controller for the premium is obtained using advanced optimization techniques, and it is finally shown that the respective pricing strategy follows a more balanced development compared with the traditional premium approaches.
 Shunquan Zhu Journal of Mathematical Finance (JMF) , 2016, DOI: 10.4236/jmf.2016.64037 Abstract: This paper is based on covariance and expected return, building portfolio risk optimization model. Using Genetic Algorithm and Quadratic Programming, three securities portfolio Optimization model is resolved, and we find that Genetic Algorithm having priority for Restraint Conditions is not a linear model.
 Jinwen Wu International Business Research , 2009, DOI: 10.5539/ibr.v1n2p34 Abstract: The essay makes a thorough and systematic study about a mean- maximum deviation portfolio optimization model. First, we make a careful analysis about the problem and build a model about this kind of problem. The essay gives two kind of different and characteristic solutions—linear programming solution and critical line solution.
 Journal of Mathematical Finance (JMF) , 2017, DOI: 10.4236/jmf.2017.73037 Abstract: This paper considers a portfolio optimization problem with delay. The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model. In addition, considering the history information related to investment performance, the dynamic of wealth is modeled by stochastic delay differential equation. The investor’s objective is to maximize her expected utility for a linear combination of the terminal wealth and the average performance. By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closed-form expressions of optimal strategy and optimal value function for CRRA utility are derived. Finally, a numerical example is provided to show our results.
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