Product expansion for stochastic jump diffusions and its application to numerical approximation

We derive a product expansion of the exponential Lie series in terms of a Philip Hall basis for the Chen series corresponding to the stochastic jump diffusion as in Sussmann (in: C.I. Byrnes and A. Lindquist (Eds.), Theory and Applications of Nonlinear Control Systems, North-Holland, Amsterdam, 1986, pp. 323–335) for the deterministic case. Based on the expansion, we establish the Stratonovich–Taylor–Hall (STH) schemes such that each scheme involves only the minimum number of multiple stochastic integrals, which can be regarded as systems of stochastic differential equations and approximated by a lower order scheme with an appropriate step size to ensure the necessary accuracy. Mean-square convergence of the STH schemes is shown and numerical examples are provided to illustrate the results.