Weak approximation for stochastic differential equations with jumps by iteration and hard bounds

We establish a novel theoretical framework in which weak approximation can be conducted in an iterative and convergent manner for a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent intensity. The proposed iteration scheme is built on a sequence of approximate solutions, each of which makes use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed iteration scheme is convergent and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of the approximate solution at each iteration step, we construct upper and lower bounding functions that are convergent towards the true solution as the iterations proceed. We provide illustrative examples so as to examine our theoretical findings and demonstrate the effectiveness of the proposed theoretical framework and the resulting iterative weak approximation scheme.