Differentiation of the Choquet Integral and Its Application to Long-term Debt Ratings

Abstract

Differentiation of the Choquet integral of a nonnegative measurable function with respect to a fuzzy measure on fuzzy measure space is proposed and it is applied to the capital investment decision making problem by Kaino and Hirota. In this paper, differentiation of the Choquet integral of a nonnegative measurable function is extended to differentiation of the Sipos Choquet integral of a measurable function and its properties will be discussed. First, the real interval limited Schmeidler Choquet integral and Sipos Choquet integral are defined for preparation, then the upper differential coefficient, the lower differential coefficient, the differential coefficient, and the derived function of the Choquet integral along the range of an integrated function are defined by the limitation process of the interval limited Choquet integral. Two examples are given, where the measurable functions are either a simple function or a triangular function. Basic properties of differentiation about swn and multiple with constant, addition, subtraction, multiplication, and division are shown. Then, the Choquet integral is applied to long-term debt ratings model, where the input is qualitative and quantitative data of corporations, and the output is Moody\'s long-term debt ratings. The fuzzy measure, which is given as the importance of each qualitative and quantitative data, is derived from a neural net method. Moreover, differentiation of the Choquet integral is applied to the long-term debt ratings, where this differentiation indicates how much evaluation of each specification influence to the rating of the corporation.