An optimization approach to weak approximation of stochastic differential equations with jumps

We propose an optimization approach to weak approximation of stochastic differential equations with jumps. A mathematical programming technique is employed to obtain numerically upper and lower bound estimates of the expectation of interest, where the optimization procedure ends up with a polynomial programming. A major advantage of our approach is that we do not need to simulate sample paths of jump processes, for which few practical simulation techniques exist. We provide numerical results of moment estimations for Doléans–Dade stochastic exponential, truncated stable Lévy processes and Ornstein–Uhlenbeck-type processes to illustrate that our method is able to capture very well the distributional characteristics of stochastic differential equations with jumps.