Luciano Stefanini

Generalized differentiability of fuzzy-valued functions

In the present paper, using novel generalizations of the Hukuhara difference for fuzzy sets, we introduce and study new generalized differentiability concepts for fuzzy valued functions. Several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them. Characterization and relatively simple expressions are provided for the new derivatives.

A generalization of Hukuhara difference and division for interval and fuzzy arithmetic

We propose a generalization of the Hukuhara difference. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers. Applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown.

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α-cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α-cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

Synchronization, intermittency and critical curves in a duopoly game

The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x1=x2, so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.

Approximate fuzzy arithmetic operations using monotonic interpolations

Abstract We suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number (or interval) and to derive a procedure to control the absolute error associated to the arithmetic operations ( + ,-, · , : ) between fuzzy numbers, in order to reduce the distance between the true result of the operation and its approximation. The monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic. Several computational experiments are given to show the good performance of the proposed procedure.

F-transform with parametric generalized fuzzy partitions

The fuzzy transform (F-transform) has some interesting properties that allow its use in curve fitting and smoothing. We suggest the following: (a) a generalization of fuzzy partitions for the F-transform to control and improve the smoothing effect and (b) a parametrization of the fuzzy numbers constituting the basic functions related to the fuzzy partitions. An optimization criterion is then suggested to obtain the F-transform with best approximation properties, by estimating the basic functions from a family of parametric fuzzy numbers. The generalized fuzzy partitions are characterized by the number of subintervals (the bandwidth of the partition) forming the core of each basic function and we suggest two procedures for bandwidth selection: one based on taut-string variation and a second based on generalized cross validation. Some illustrative examples are included.

A Generalization of Hukuhara Difference

We propose a generalization of the Hukuhara difference. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is seggested to attempt a generalization of division for real intervals and fuzzy numbers.

Option price sensitivities through fuzzy numbers

The main motivation in using fuzzy numbers in finance lies in the need for modelling the uncertainty and vagueness that are implicit in many situations. However, the fuzzy approach should not be considered as a substitute for the probabilistic approach but rather as a complementary way to describe the model peculiarities. Here, we consider, in particular, the Black and Scholes model for option pricing, and we show that the fuzzification of some key parameters enables a sensitivity analysis of the option price with respect to the risk-free interest rate, the final value of the underlying stock price, the volatility, and also better forecasts (see Thavaneswaran et al. (2009) [12] for details). The sensitivities with respect to the variables of the model are represented by different letters of the Greek alphabet and they play an important role in the definition of the shape of the fuzzy option price.

Interval and fuzzy Average Internal Rate of Return for investment appraisal

In investment appraisal, uncertainty can be managed through intervals or fuzzy numbers because the arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. We apply interval and fuzzy numbers to the Average Internal Rate of Return (AIRR), recently introduced for overcoming the problems of the traditional Internal Rate of Return (IRR). In the setting of interval and fuzzy arithmetic, we establish relations between the interim capitals invested, the profits and the cash flows, which are the ingredients of the AIRR and shed lights on the different ways uncertainty propagates depending on which variable is known and which one is derived. The relations between fuzzy AIRR and fuzzy Net Present Value are also investigated.

Solution of Fuzzy Differential Equations with generalized differentiability using LU-parametric representation

The paper uses the LU-parametric representation of fuzzy numbers and fuzzy-valued functions, to obtain valid approxi- mations of fuzzy generalized derivative and to solve fuzzy differential equations. The main result is that a fuzzy differential initial-value problem can be translated into a system of innitely many ordinary differential equations and, by the LU-parametric representation, the innitely many equations can be approximated efciently by anite set of four ODEs. Some examples are included.