Ergodicity For A Class Of Hybrid Stable Processes And Applications

Stochastic differential equations have been widely used in many fields,among which Ginzburg-Landau equation has been widely used in the field of physics since it was proposed.This paper investigates the Ginzburg-Landau equation with Markov switch driven by ?-stable processes.The stability and ergodicity of the solution of the hybrid pure jump processes are discussed by the Lyapunov method and the Poisson equation method.For any ??(1,2),we give a sufficient condition for the following equation to be asymptotically stable in probability at equilibrium Xt?0.dXt=(artXt-brtXt3)dt+?rtXtdZt.For any ??(1,2),We give a sufficient condition for the following equation to be ergodic and to have a unique stationary distribution.dXt=(artXt-brtXt3)dt+?rt+dZt.In addition,the parameter estimation algorithm for the hybrid Ginzburg-Landau equations are presented with ?=2.Numerical simulations are given to verify the accuracy of the parameter estimation.We use such equations to model the Shanghai Composite Index from January 4,2012 to December 31,2021.