utility function

This study introduces the Choquet integral of fuzzifying functions with respect to a fuzzy measure. To express various phenomena or ambiguous values in many applications may not be enough to show them as a function, in which case a fuzzifying function can be applied to achieve better expressions or flexibility for given function values. To apply Choquet integrals of fuzzifying functions, we consider Choquet integrals of interval-valued functions as an operator which are α-level functions of fuzzifying functions. In this study, we investigate some properties of Choquet integral of fuzzifying functions and present their applications. As part of this, a series of relevant examples and their subsequent applications are provided, along with the fuzzification of two integrands: a probability density function (PDF) and a utility function.

In this article, an outer measure is constructed on a pseudoordered set (X, ≿) and then it will be shown that it is in fact a measure defined on the whole power set of X . Applying this, a measurable utility function θ is defined which represents the relation ≿ on X. Also, we discuss the continuity and upper semi-continuity of θ in certain points of X. Finally, the results are used to improve some of the theorems in economics.

The nonlinear Black-Scholes equation has been increasingly attracting interest over the last two decades, because it provides more accurate values by considering transaction costs as a viable assumption. In this paper we review the fully nonlinear Black-Scholes equation with an adjusted volatility which is a function of the second derivative of the price and then we prove two new theorems in this realistic model.