Initial Value Estimation of Uncertain Differential Equations Based on Residuals with Application in Financial Market

The initial value estimation of uncertain differential equations refers to the process of estimating the initial state of a time-varying system using observed data when we cannot know exactly the initial state of the system in uncertain environments. In order to study the initial value estimation problem of uncertain differential equations, this paper constructs the confidence interval and point estimation of the initial value based on the residuals corresponding to the observed data. In order to further explain the initial value estimation based on residuals, this paper gives the confidence intervals and point estimations of the initial value for several specific uncertain differential equations, including linear, exponential, and mean-reversion uncertain differential equations. Finally, two numerical examples and an empirical study are also provided to illustrate the effectiveness of the proposed method.