Properties Of Solutions Of Interest Rates Mean Reversion θ Model With State-dependent Switching

Recently, with the rapid development of financial mathematics, stochastic differential equations have been widely applied in financial community. However, in financial markets, it is well-known that short-term interest rates are one of the most fundamental and important financial quantities. Nowadays, there have been many scholars who have studied the short-term interest rate with a variety of classical differential equation models, where many outstanding results are successfully obtained. Considering the description of short-term interest rates with many classical differential equations, in this paper, we mainly study interest rates mean reversion q- process which is of strong representativeness. As a special case of this model, the Cox-Ingersoll-Ross interest rate model is a very classic model, which has been systematically investigated in many differential equations. For example, in Zhang Nan’s paper, the author studied some properties of the solutions of the CIR model with continuous-state-dependent switching. In this paper, based on [1], some properties of the solutions with state-dependent switching are extended to more general interest rates mean reversion q- process, where θ∈(1,0].Because of the complexity of the model and unavailability of its exact solutions, we borrow Lipschitz continuity and localized analysis to deal with the difficulty of state dependence and employ the maximization to overcome the problem caused by qparameter in the proof of our theories. There are four main parts as follows:Firstly, we prove that interest rates mean reversion q- process with continuous-state-dependent switching have a global positive or non-negative solutions.Secondly, continuity, smoothness and Feller continuity dependent on initial data are demonstrated. In addition, we prove that certain functions of the solutions satisfy a system of Kolmogorov’s back-ward differential equations. Finally, the error estimate of the numerical solutions of interest rates mean reversion q model is given.