The Properties Of Mean Reversion θ Model’s Solutions With State-dependent Regime Switching

The birth of the stochastic differential equations has its certain application background. In the 70’s and 80’s, stochastic differential equations, diffusion process and random analysis with their rapid development are widely employed in system science, engineering science, finance and other aspects, and there successively appeared Gihan-Skorohod, Ikedawatake and other famous scholars. In recent years, the use of equations concept of random financial qualitative and quantitative analysis is becoming more common, such as the application of martingale and random points to describe the market and to calculate the option price. With the development of economy, the phenomenon of various financial and economic market pattern are under all kinds of uncertain dynamic factors and some complex changes. Stochastic differential equation theory to describe the nature of complex financial phenomenon also in deepening degree, By geometric Brownian motion to the classic CIR model to CIR model with continuous state dependent switch, This model has good properties, such as the true solution have weak continuity and so on. This article with continuous state dependent switch CIR model generalization of promotion, get mean reversion θ model with continuous state dependent switch,Among them θ∈ [0.5,1]. The switch function not only related to the state of t but aslo related to the moment of asset prices S (t), that is:So it makes this problem difficult. Sign the characteristics in this kind of problem, we will consider two processes (S(t),α(t)) at the same time and the definition of associated will be given.This article, we mainly study some properties of such equations, first we will give the existence and uniqueness of solution, weak convergence, ergodicity, numerical solution and the numeric approximation of the real solution will be given. In the process of proof necessary assumptions and definitions also are given.