The Least Squares Estimation of Uncertain Fractional Differential Equations with Application to the Financial Markets

In uncertain dynamical systems, there is a class of parameter estimation problems for fractional differential equations (UFDEs) that are important and related research is scarce. Given this, the parameter estimation problem for Caputo type of UFDEs under the initial conditions is proposed. The least squares estimation method is a method that minimizes the sum of squared errors between the obtained data and the actual data. The estimation of unknown parameters in the UFDEs are obtained by the least squares estimation method based on limited observed values, and the rationality of the parameters is tested following uncertain hypothesis testing. A specific algorithm for parameter estimation is provided, and the effectiveness of the proposed model is demonstrated by two examples in the financial markets: Alibaba's stock price and the exchange rate between the RMB and Euro.