The Schr\"odinger Bridge Problem for Jump Diffusions with Regime Switching

TL;DR

This paper studies the Schr"odinger bridge problem for jump diffusions with regime switching, providing a comprehensive analysis of the optimal measure and its properties in this novel setting.

Contribution

It introduces the first analysis of the SBP for regime-switching jump diffusions, establishing properties of the optimal measure and applying the theory to unbalanced SBP cases.

Findings

The optimal measure remains a regime-switching jump diffusion.

Established stochastic calculus and analytic properties of the solution.

Provided new insights into unbalanced SBP within this framework.

Abstract

The Schr\"odinger bridge problem (SBP) aims at finding the measure $\hat{\mathbf{P}}$ on a certain path space which possesses the desired state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence from a reference path measure $\mathbf{R}$. This work focuses on the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion with regime switching, which is a Markov process that combines the dynamics of a jump diffusion with interspersed discrete events representing changing environmental states. To the best of our knowledge, the SBP in such a setting has not been previously studied. In this paper, we conduct a comprehensive analysis of the dynamics of the SBP solution $\hat{\mathbf{P}}$ in the regime-switching jump-diffusion setting. In particular, we show that $\hat{\mathbf{P}}$ is again a path measure of a regime-switching jump diffusion; under proper assumptions, we establish various properties of $\hat{\mathbf{P}}$ from both a stochastic calculus perspective and an analytic viewpoint. In addition, as an demonstration of the general theory developed in this work, we examine a concrete unbalanced SBP (uSBP) from the angle of a regime-switching SBP, where we also obtain novel results in the realm of uSBP.