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.

翻译：我们分析了与两个多维跳跃-扩散过程相关的两个概率分布之间的Wasserstein距离（$W$距离）。具体而言，我们分析了一种时间解耦的平方$W_2$距离，该距离为两个跳跃-扩散过程之间漂移函数、扩散函数和跳跃幅度函数的差异提供了上下界。在此基础上，我们提出了一种时间解耦的平方$W_2$距离方法，通过参数化神经网络从数据中高效重构未知的跳跃-扩散过程。我们进一步证明，利用跳跃-扩散过程漂移函数的先验信息可以提升该方法的性能。多个示例与应用验证了我们所提重构方法的有效性。