Parametric analysis of the oscillatory solutions to SDEs with Wiener and Poisson components by a Monte Carlo method

We investigate the influence of Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs) with the use of a Monte Carlo method. For linear and Van der Pol oscillators, we investigate the accuracy of the estimates of the functionals of numerical solutions to SDEs obtained by the generalized Euler explicit method. For the linear oscillator, the exact analytical expressions of the mathematical expectation and the variance of the solution to the SDE are obtained. These expressions allow us to investigate the dependence of the accuracy of the estimates of the moments of the solution on the values of the parameters of the SDE, the size of the integration step, and the size of the ensemble of the simulated trajectories of the solution. For the Van der Pol oscillator, the dependence of the frequency and the decay rate of the oscillations of the mathematical expectation of solution to the SDE on the values of the parameters of the Poisson component is investigated. The results of numerical experiments are presented.