Unified Numerical Method for Stochastic Differential Equations with Poisson and Gaussian White Noises

A method is developed for integrating stochastic differential equations (SDEs) with Poisson (PWN) and Gaussian (GWN) white noises interpreted as the formal derivatives of the compound Poisson and Brownian motion processes. In contrast to the current integration schemes, which solve discrete time versions of the posed SDEs, the proposed method solves the posed SDEs for finite dimensional (FD) models of the compound Poisson and Brownian motion processes, i.e., finite sums of deterministic functions of time weighted by random coefficients. Paths of the resulting solutions, referred to as FD solutions, can be generated by standard ordinary differential equation (ODE) solvers since the paths of the FD input models are smooth. We also establish conditions under which the distributions of extremes and other continuous functionals of the solutions of the posed SDEs can be approximated by those of their FD solutions. This is essential in applications since the distributions of functionals of FD solutions can be estimated while those of actual solutions are rarely available analytically and cannot be obtained numerically.