The estimation of probability distribution of SDE by only one sample trajectory

Abstract

This paper focuses on the method of the simulation of a stochastic system and the main method of our paper is the Monte Carlo computation simulation method. Taking the stochastic Logistic equation as an example, we present the simulation of the sample trajectory by Euler scheme and the invariant probability distribution of stochastic differential equations with the Monte Carlo method. We also compare the simulation result with the analytical result for the autonomous stochastic Logistic model. Moreover, the stochastic Logistic equation with Markovian switching which is described by a Markov chain taking values in a finite state space is considered.