Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.