A Regime-Switching Approach to the Unbalanced Schr\"odinger Bridge Problem

TL;DR

This paper introduces a regime-switching diffusion framework to solve the unbalanced Schr"odinger bridge problem, accommodating various constraints on killing times and locations, and providing a unified approach that extends prior methods.

Contribution

It proposes a novel regime-switching diffusion approach for uSBP, analyzing different constraints and establishing connections among them, thereby extending existing literature.

Findings

Analyzed four different constraint scenarios for uSBP.

Established rigorous connections among different uSBP cases.

Provided a unified framework extending previous approaches.

Abstract

The unbalanced Schr\"odinger bridge problem (uSBP) seeks to interpolate between a probability measure $\rho_0$ and a sub-probability measure $\rho_T$ while minimizing KL divergence to a reference measure $\mathbf{R}$ on a path space. In this work, we investigate the case where $\mathbf{R}$ is the path measure of a diffusion process with killing, which we interpret as a regime-switching diffusion. In addition to matching the initial and terminal distributions of trajectories that survive up to time $T$, we consider a general constraint $\psi(t,x)$ on the distribution of killing times and/or killing locations. We investigate the uSBPs corresponding to four choices of $\psi$ in detail which reflect different levels of information available to an observer. We also provide a rigorous analysis of the connections and the comparisons among the outcomes of these four cases. Our results are novel in the field of uSBP. The regime-switching approach proposed in this work provides a unified framework for tackling different uSBP scenarios, which not only reconciles but also extends the existing literature on uSBP.