Extreme values for solution to uncertain fractional differential equation and application to American option pricing model

Abstract

Uncertain fractional differential equation plays an important role of describing uncertain dynamic process. This paper focuses on extreme values (including supremum and infimum) for solution to an uncertain fractional differential equation for the Caputo type. Theorems for the inverse uncertain distributions of the extreme values are given based on the definition of α-path. And then, numerical algorithms for them are designed, numerical examples are shown for validating the availability about algorithms. The absolute errors between the numerical and analytical results are also presented to demonstrate the accuracy of the algorithms. Finally, as an application of the extreme values, an uncertain stock model is proposed on the basis of uncertain fractional differential equation of the Caputo type. The American option pricing formulas of such stock model are studied by using the proposed extreme theorems. Besides, numerical calculations are also illustrated with respect to different parameters p.