Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model

Abstract

Uncertain fractional order differential equation is a significant tool for modeling the uncertain dynamic system. First, we consider solutions of an uncertain fractional order differential equation with the Caputo type and investigate inverse uncertain distributions of their time integral. On the basis of α-path, two different time integral theorems for inverse uncertain distributions are given. Second, in uncertain financial markets, the interest rate is considered as an uncertain process. As the application of the time integral, we present a novel zero-coupon bond model and derive a pricing formula of zero-coupon bond under this model. Last, by the predictor-corrector method, the numerical algorithm is designed. Analytic expressions and numerical calculations of the zero-coupon bond price are illustrated for fractional order mean-reverting model and standard deviation model, respectively.