Long Time Behavior For A Class Of Hybrid Nonlinear Systems And Their Applications

With the continuous expansion of the scale of our country's stock market,the interaction between the stock market and the macro economy has become more and more significant.Therefore,stock price prediction and other related issues have received extensive attention from many scholars.A large number of financial empirical studies have shown that stock prices are characterized by nonlinearity and fat-tailedness.Many studies also shown that most financial risk problems often exist in the fat tail.In addition,stock prices are susceptible to a variety of complex factors that cause sudden price changes,resulting in jumping phenomenon.In order to remedy for the inapplicability of Gaussian process,the ?--stable processes can well describe the phenomenon of fat-tailedness and jumps,and the markovian switching is used to fit the bull-bear alternation of the stock market.We discuss a class of Ginzburg–Landau type hybrid nonlinear systems(GL type systems for short)driven by symmetric ?--stable processes with markovian switching for fitting.In the theoretical research,we mainly study the long-term behavior of GL-type systems.First,we provid the su cient conditions for the ergodicity of the GL type systems.To this end we prove the existence of the stationary distribution of the system by the maximum principle and use Khasminskii's lemma and Lyapunov method to prove that the system has a unique stationary distribution.Secondly,by using the M-matrix and Gaussian hypergeometric distribution,we obtain the su cient conditions for transience of the GL type systems.In the empirical study we based on the above theory.At first we take the closing prices of Shanghai Pudong Development Bank in the past ten years as a sample,and use stable distribution for fitting.The four parameters of the stable distribution are obtained by the quantile method and verified by the Pearson test.After comparison,it can be seen that the fitting e?-ect of the stable distribution is better than that of the normal distribution.Finally,we use the computer iterative method to make the path graph of the GL-type systems under di?-erent ?- values.It has been observed that when ?- is closer to 2,the process is more stable.We also use the expectation-maximization algorithm to obtain the estimator of the GL-type equation parameters.