Portfolio Optimization Method Based On Equilibrium Theory

The equilibrium measure is a natural extension of both probability and cred-ibility measures. The convergence mode of random fuzzy variables with respectto equilibrium measure is an important issue for research. In this thesis, we firstintroduce several convergence concepts for sequences of random fuzzy variables,including convergence in equilibrium measure and convergence in equilibrium dis-tribution. Then we deal with the properties of the convergence modes of randomfuzzy variables. The equilibrium mean value of random fuzzy variable with respectto equilibrium measure is also defined by nonlinear integral. For sequence of inte-grable random fuzzy variables, we deal with the important monotone convergencetheorems as well as dominated convergence theorems.Based on the equilibrium theory, we propose a new portfolio optimization modelin hybrid uncertain decision systems. In this thesis, EV-VaR model takes the ex-pected value of the return as the objective function, and uses value-at-risk measureto construct constrain. Considering the uncertainties of actual investment activities,we model the return rate by a random fuzzy variable. In order to solve this problem,we combine probability and credibility measures, and discuss the properties of modelin general case, and derive a determined convex programming model. In the casethat the expected value of the return rate are triangular, trapezoidal and normallyfuzzy variables, we give the equivalent deterministic programming model. Throughthe numerical experiments, we obtain the relationship between EV and VaR. Wealso carry out the sensitivity analysis by changing the values of parameters. Thecomparative study about the model in hybrid uncertain decision systems and themodel in random system indicates the superiority of the proposed new optimizationmethod.The major new results of this thesis contain the following four aspects:(i) Wedevelop the convergence modes of random fuzzy variables in equilibrium theory,and establish their interconnections;(ii) We introduce the equilibrium mean valueof random fuzzy variable, and establish the important monotone convergence theo- rems and dominated convergence theorems for sequences of integrable random fuzzyvariables;(iii) Based on equilibrium theory, we propose a new portfolio equilibriumoptimization method, establish the EV-VaR model, analyze the properties of modeland design solution method;(iv) Through the numerical experiments, we illustratethe feasibility and efectiveness of the proposed optimization method.