Generalized Nash Equilibrium Problems And Pricing Currency Option Under Fuzzy Environment

The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP) by allowing that both the objective function and the strategy set of each player depend on the rival players’strategies. So, the GNEP is more suitable for describing the actual competitive market. However, the numerical algorithms are extremely scare. In addition, the generalized Nash equilibrium problem with general cone constraint seems a blank in the studies. In this thesis, we investigate two penalty methods for solving the generalized Nash equilibrium problem, and Nash equilibrium problem with semidefinite cone constraints. And we discuss pricing currency option problem under fuzzy environment. With the fast-growing of the trading volume in the foreign exchange market, the trading of currency option has increased. It is well known that currency option manages the risk of the foreign exchange market. In general, the data cannot be recorded or collected precisely. So, we assume market date as fuzzy numbers and the discussion for pricing currency option problem under fuzzy environment is rational. Specifically, the main contributions are summarized as follows:1. In Chapter3, we discuss the equivalence relation between the GNEP and quasi-variational inequality problem. Then, by means of the exponential and smoothing y norm penalty functions, we propose two new sequential penalty approaches for solving general GNEPs, where each penalized problem of the sequence is a C2smoothing penalized NEP. Under the EMFCQ at a limit point of solutions to smoothing NEPs, we demonstrate that the limit point is a solution to the GNEP. We formulate the Karush-Kuhn-Tucker (KKT) conditions for the NEP into a system of nonsmooth equations, and then apply the semismooth Newton method with Armijo line search to solve the system. Finally, the numerical results show that our two penalty methods for GNEPs are effective.2. In Chapter4, we consider an inexact Newton method for Nash equilibrium problem with semidefinite cone constraints. Firstly, the Karush-Kunh-Tucker system of the Nash equi-librium problem with semidefinite cone constraints is formulated into a system of non-smooth equations with help of the matrix-valued natural residual function. Under some conditions, we prove that the Clarke generalized Jacobian of the system of nonsmooth equations in solution points is nonsingular. To the end, we propose an inexact Newton method to the system of nonsmooth equations. 3. In Chapter5, we consider pricing currency option problem under fuzzy environment. Firstly, we modify the definition of fuzzy number by another definitions of the0-level set and the support set of fuzzy subset. And we obtain a basic proposition about the fuzzy-valued functions of fuzzy subsets. Based on that basic proposition and the extension principle, we prove that the fuzzy price obtained from the Garman-Kohlhagen formula for European currency option is a fuzzy number. Finally, we provide an defuzzification method based on identification weighting parameter, which can identify the optimal pa-rameter.