Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model

Uncertain differential equation is a type of differential equation driven by Liu process. Compared to one dimensional uncertain differential equations, more situations can be considered by multi-dimension uncertain differential equations to accurately describe multiple risk processes. In this manuscript, a special type of high dimensional uncertain differential equations, named nested uncertain differential equations, are introduced. The α-path of the proposed nested uncertain differential equation will be derived. Furthermore, a numerical method is designed for solving nested uncertain differential equations using the proposed α-path method. We also used nested uncertain differential equations to build a multi-factor term structure model for short interest rate where the drift is described by another uncertain differential equation. The pricing of the zero-coupon bond will calculated based on the proposed model.