First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model

Abstract

Uncertain fractional differential equation with memory and hereditary characteristics is a useful way to better model the uncertain dynamic system. Firstly, the solution of an uncertain fractional differential equation with the Caputo type is considered and uncertain distributions of their first hitting time is investigated. On the basis of the α-path, two different first hitting time theorems for uncertain distributions are proposed. Secondly, by the predictor-corrector method, the numerical method is designed. A nonlinear example is provided for validating the availability of the proposed algorithm. Then, as an application of the first hitting time, a novel uncertain risk index model is presented and a formula of risk index under our model is derived accordingly. Lastly, the numerical algorithm of risk index is designed and numerical calculations for the risk index are illustrated with regard to different parameters.