The Schrödinger Bridge Problem for Jump Diffusions with Regime Switching
Andrei Zlotchevski, Linan Chen
TL;DR
This work extends the Schrödinger bridge problem (SBP) to regime-switching jump diffusions by formulating SBP on the hybrid state space $\mathbb{R}^d\times\mathcal{S}$ and minimizing $\mathrm{KL}(\mathbf{P}\|\mathbf{R})$ subject to endpoint laws $\rho_0,\rho_T$. It proves that the SBP bridge $\widehat{\mathbf{P}}$ remains a regime-switching jump diffusion, obtained via a Girsanov transform driven by a harmonic Schrödinger potential $\varphi$, and it connects this bridge to a stochastic control problem with optimal controls $(u^*,\theta^*,\xi^*)$. When a transition density exists, the paper derives forward and backward PIDEs for the Schrödinger potentials and provides a regime-switching Fortet–Sinkhorn algorithm for computation. As a concrete instantiation, it analyzes the unbalanced SBP (uSBP) for jump diffusions with killing, yielding explicit forms for the optimal controls and the bridge dynamics, including how mass loss is distributed across active and dead regimes. Overall, the results generalize SBP theory to regime-switching jump diffusions, enabling modeling of environmental regime changes and mass-loss phenomena within entropic optimal transport frameworks.
Abstract
The Schrödinger bridge problem (SBP) aims at finding the measure $\hat{\mathbf{P}}$ on a certain path space which possesses the desired state-space distributions $ρ_0$ at time $0$ and $ρ_T$ 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.