We establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of L\'evy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making 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 recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.

翻译：针对一类具有广义时间状态相依无界跳强度（非列维型驱动，后者因独立平稳增量而具有显著优势）的多变量非齐次随机微分方程，我们建立了一种完全解耦跳过程的递归表示。该递归表示及其若干相关形式，通过将动力学系统的跳时刻作为信息中继点，将历史信息传递至前一迭代步骤以填补未观测轨迹的缺失信息而导出。我们证明所提出的递归表示在极限下呈指数快速收敛，并且可在抑制跳分量的概率测度下表示为类似皮卡迭代的形式。基于每次迭代结果，我们构造了上下界函数，这些函数随着迭代进行同样收敛至真实解。数值实验结果验证了理论发现。