The Schr\"odinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Under mild assumptions, with both the operator theory approach and the stochastic calculus techniques, we establish an $h$-transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution $\hat{\mathbf{P}}$ as the strong-convergence limit of a sequence of harmonic $h$-transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the $h$-transform framework and the approximation method developed in this work are robust and applicable to a relatively general class of jump diffusions. In addition, we examine the SBP of particular types of jump diffusions under additional regularity conditions and extend the existing results on the SBP from the diffusion case to the jump-diffusion setting.

翻译：薛定谔桥问题（SBP）旨在寻找特定路径空间上的测度 $\hat{\mathbf{P}}$，该测度在时间 $0$ 的状态空间分布 $\rho_0$ 与时间 $T$ 的分布 $\rho_T$ 之间进行插值，同时最小化其与参考路径测度 $\mathbf{R}$ 的KL散度（相对熵）。本文中，我们处理当 $\mathbf{R}$ 为跳跃扩散路径测度时的SBP。在温和假设下，通过算子理论方法与随机分析技术，我们建立了跳跃扩散的 $h$-变换理论，并设计了一种逼近方法，通过一系列调和 $h$-变换的强收敛极限来实现跳跃扩散SBP的解 $\hat{\mathbf{P}}$。据我们所知，这些结果在SBP研究中具有新颖性。此外，本文发展的 $h$-变换框架与逼近方法具有鲁棒性，可适用于相对广泛的一类跳跃扩散。另外，我们在附加正则性条件下研究了特定类型跳跃扩散的SBP，并将现有关于SBP的结果从扩散情形推广至跳跃扩散场景。