Invariant distribution of stochastic Gompertz equation under regime switching

Abstract

This paper is concerned with the asymptotic behaviors of a stochastic Gompertz model in random environments from the view of Itô stochastic differential equations with Markovian switching. Based upon the deterministic Gompertz model, we establish the corresponding stochastic model which is described as a stochastic Gompertz models with Markovian switching. We show that this model is asymptotically stable in distribution and that it displays an invariant probability distribution under certain conditions. Most importantly, we simulate the trajectories and the limits probability distribution of the solution with the method of Monte Carlo stochastic simulation. The simulation results illustrate that our conclusions are correct, and moreover the results reflect the statistical properties of the stochastic model.