Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market

Uncertain fractional differential equations (UFDEs) have non-locality features to reflect memory and hereditary characteristics for the asset price changes, thus are more suitable to model the real financial market. Based on this characteristic, this paper primarily investigates the monotonicity theorem for uncertain fractional differential equations in Caputo sense and its application. Firstly, monotonicity theorems for solutions of UFDEs are presented by using the a-path method. Secondly, as the application of the monotone function theorem, a novel uncertain fractional mean-reverting model with a floating interest rate is presented. Lastly, the pricing formulas of the European and American options are derived for the proposed model based on the monotone function and present extreme values and time integral theorems, respectively. In addition, numerical schemes are designed, and numerical calculations are illustrated concerning different parameters through the predictor-corrector method. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

This paper investigates the application of uncertain fractional differential equations in financial market modeling, presenting monotonicity theorems and an uncertain fractional mean-reverting model. Pricing formulas for European and American options are derived based on monotone function and present extreme values and time integral theorems. Numerical schemes are designed and illustrated for different parameters through the predictor-corrector method.