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 α-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 α∈（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+σrtXtdZt.For any α∈（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+σrt+dZt.In addition,the parameter estimation algorithm for the hybrid Ginzburg-Landau equations are presented with α=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.