Exact solutions and doubly efficient approximations of jump-diffusion itô equations

Abstract

Abstract This paper presents exact solutions to an unsolved class of jump-diffusion stochastic differential equations and derives efficient numerical schemes for the general non-linear cases. It is proved that even the second order mean square efficient schemes may not be second order efficient in the weak sense. The generator and Taylor expansion in the expectationssemi-group are verified under weaker conditions and applied to derive new doubly efficient schemes which are proved to converge with the best possible order rate in both senses. A class of direct jump-adapted schemes are als o presented. Comparative simulations are consistent with the findings.