Runge-Kutta methods for jump-diffusion differential equations

In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.