广义纳什均衡问题与模糊环境的货币期权定价

发布时间：2018-03-11 12:19

本文选题：广义纳什均衡问题　切入点：惩罚函数方法　出处：《大连理工大学》2013年博士论文　论文类型：学位论文

【摘要】：广义纳什均衡问题(GNEP)是纳什均衡问题(NEP)的推广,它允许每个参与人的目标函数和策略集都可以依赖于竞争者的策略.故GNEP更适合于描述实际的竞争市场,但是数值算法还很少.此外,关于一般锥约束形式的GNEP还鲜有讨论.本文研究了两个求解GNEP的惩罚函数方法,以及一个半定锥约束的纳什均衡问题；另外我们还讨论了模糊环境的货币期权定价问题.随着外汇市场交易量的迅速增长,货币期权的交易量也在逐渐增加.众所周知,货币期权能够有效地管理外汇市场的风险.一般来说,数据是不能被完全精确记录和搜集的,因此我们假设市场中的数据为模糊数,考虑模糊环境的货币期权定价是合理的.具体来说,我们得到如下结果： 1.第三章中在讨论了广义纳什均衡问题与拟变分不等式之间的等价关系之后,我们利用指数型和光滑化的y范数惩罚函数提出了两个新的求解一般形式的广义纳什均衡问题的序列惩罚方法,其中序列中的每个惩罚问题都是C2光滑的惩罚纳什均衡问题.我们证明了若光滑的惩罚纳什均衡问题序列的解序列的聚点处EMFCQ成立,则此聚点是广义纳什均衡问题的一个解.进一步,我们把惩罚纳什均衡问题的Karush-Kuhn-Tucher条件转化为一个非光滑方程系统,然后再用带有Armijo线搜索的半光滑牛顿法来求解此系统.最后,数值结果表明我们的两个惩罚函数方法是有效的. 2.第四章中研究了求解半定锥约束的纳什均衡问题的非精确牛顿法.首先,利用矩阵值的自然残差函数将半定锥约束的纳什均衡问题的Karush-Kunh-Tucker系统转化为非光滑方程组.然后,证明了在一定条件下此方程组在解点处的Clarke广义Jacobian是非奇异的.最后,应用非精确牛顿法求解此方程组. 3.第五章中研究了模糊环境的货币期权定价问题.首先利用新的模糊集的0-水平集和支撑集的定义,我们修改了模糊数的定义,从而得到了一个关于模糊集的模糊值函数的基本命题.接着,我们利用扩展原理和上述基本命题,给出了一个用于欧式货币期权定价的Garman-Kohlhagen公式的模糊版本,并证明它是一个模糊数.最后,通过辨识最优权参数,我们提出了一个利用权参数辨识的解模糊化方法.
[Abstract]:The generalized Nash equilibrium problem (GNEP) is the Nash equilibrium problem (NEP) of the promotion, the objective function and the strategy allows each participant can set depends on the competitor's strategy. Therefore, GNEP is more suitable to describe the actual market competition, but also a few numerical algorithm. In addition, the general form of the GNEP cone constraints little discussion. This paper studies the penalty function method and two for GNEP, and 1.5 fixed cone constrained Nash equilibrium problem; we also discussed the issue of currency option pricing fuzzy environment. With the rapid growth of foreign exchange market trading volume, trading volume of currency options are increasing gradually. As everyone knows, the currency option risk effectively the management of the foreign exchange market. Generally speaking, the data can not be completely accurate records and collected, so we assume that the market data as a fuzzy number, considering the fuzzy environment of monetary period The price of the right is reasonable. In particular, we get the following results:
1. in the third chapter, after the discussion of the equivalence between the generalized Nash equilibrium problem and variational inequality between, we use exponential smoothing norm and Y penalty function proposed sequence of generalized Nash equilibrium problem two new solving the general form of punishment, the punishment of each sequence are punished the Nash equilibrium of smooth C2. It is proved that if the point of EMFCQ solution sequence punish Nash equilibrium problem of smooth establishment, this point is a solution of the generalized Nash equilibrium problem. Further, we converted the Karush-Kuhn-Tucher condition to punish Nash equilibrium problem as a system of nonsmooth equations, and then with Armijo line search semi smooth Newton method is used to solve this system. Finally, the numerical results show that the two penalty function of our method is effective.
Inexact Newton method for the Nash equilibrium of 2. in the fourth chapter of solving semidefinite cone constraints. Firstly, the Karush-Kunh-Tucker system of the Nash equilibrium problem of natural residual function using matrix valued semidefinite cone constraints into nonsmooth equations. Then, it is proved that under certain conditions this equation is non singular in the Clarke generalized Jacobian solution point. Finally, the application of inexact Newton method for solving the equations.
3. the fifth chapter studies the problem of currency option pricing environment. Firstly, fuzzy definition of new fuzzy set 0- level sets and support sets, we modify the definition of fuzzy number, thus obtained the basic proposition of a fuzzy set of fuzzy valued function. Then, we use the extended principle and the basic proposition that gives a fuzzy version of a currency used for European option pricing formula of Garman-Kohlhagen, and it is proved that a fuzzy number. Finally, through the identification of optimal weight parameters, we propose a right to use the parameter identification of defuzzification method.

【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F830.9;F224

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