We analyze the Wasserstein distance ($W$-distance) between two probability
distributions associated with two multidimensional jump-diffusion processes.
Specifically, we analyze a temporally decoupled squared $W_2$-distance, which
provides both upper and lower bounds associated with the discrepancies in the
drift, diffusion, and jump amplitude functions between the two jump-diffusion
processes. Then, we propose a temporally decoupled squared $W_2$-distance
method for efficiently reconstructing unknown jump-diffusion processes from
data using parameterized neural networks. We further show its performance can
be enhanced by utilizing prior information on the drift function of the
jump-diffusion process. The effectiveness of our proposed reconstruction method
is demonstrated across several examples and applications.

通过分析两个多维跳扩散过程的概率分布之间的 Wasserstein 距离（W - 距离），我们提出了一种时间解耦的平方 W_2 - 距离方法，该方法提供了与两个跳扩散过程的漂移、扩散和跳跃幅度函数之间的差异相关的上下界。然后，我们利用参数化神经网络从数据中高效重建未知的跳扩散过程，并展示了通过利用跳扩散过程的漂移函数的先验信息可以提高其性能。我们还通过几个示例和应用程序演示了我们提出的重建方法的有效性。