Decision criteria on fuzzy variables for portfolio selection with fuzzy returns.

Abstract

In the literature, three measures were proposed to deal with imprecision and uncertainty in phenomenons. Zadeh [40] proposed the two first measures, namely possibility and necessity measures, and they enable to determine and study fuzzy variable’s parameters and to apply theoretical results in some research areas (medical diagnosis, robot control, strategic decision, games,...). The third measure, namely credibility measure and proposed by Liu [20], is a dual measure and the average of the two first measures. Following that, scholars (Liu [20], Huang [11], Li et al. [16]) determined the three first moments (mean, variance, skewness) and the first semi-moment (semi-variance) of a fuzzy variable. They used the obtained results to solve portfolio selection problem with fuzzy returns by means of the mean-semi-variance model and the mean-variance-skewness model. Furthermore, Peng et al. [27] introduced two dominance relations on fuzzy variables, namely the first and the second order dominances, and they used them to analyze risk in fuzzy context. In this thesis, we introduce moments and semi-moments of order k (k ∈ N) of a fuzzy variable and we study their properties. We introduce a new dominance relation on fuzzy variables, we characterize three dominance relations (the two previous ones and the new one) and determine their properties. TASSAK DEFFO CHRISTIAN PhD. Thesis 1 The obtained theoretical results are applied to solve the main problem of portfolio selection with fuzzy returns described by triangular fuzzy numbers by means of two approaches: the first approach based on four first moments (mean, variance, skewness, kurtosis) and the second approach based on the core of portfolios of a finite family of assets, that is, the subset of non dominated portfolios