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Schrödinger Bridge Problem for Jump Diffusions

Andrei Zlotchevski, Linan Chen

TL;DR

This work extends Schrödinger bridge analysis from diffusions to jump diffusions by formulating SBP as a KL minimization relative to a jump-diffusion path measure $\mathbf{R}$. It develops a robust $h$-transform framework for jump diffusions, proving existence and characterizing the SBP solution $\hat{\mathbf{P}}$ as either a harmonic $h$-transform or a strong limit of harmonic $h$-transforms, under mild assumptions. The paper also derives the SBP in density regimes, obtaining a forward–backward Schrödinger system with $\varphi$ and $\hat{\varphi}$ that determine the marginal dynamics, and provides explicit solvable cases (e.g., delta initial condition and Poisson) and a mollification-based approach for stable-like jump components. These results connect SBP with nonlocal operators, Lévy-type dynamics, and stochastic control, offering practical tools for inferring jump-diffusion bridges and enabling OT-inspired interpolation in jump-diffusion models.

Abstract

The Schrödinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $ρ_0$ at time $0$ and $ρ_T$ 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.

Schrödinger Bridge Problem for Jump Diffusions

TL;DR

This work extends Schrödinger bridge analysis from diffusions to jump diffusions by formulating SBP as a KL minimization relative to a jump-diffusion path measure . It develops a robust -transform framework for jump diffusions, proving existence and characterizing the SBP solution as either a harmonic -transform or a strong limit of harmonic -transforms, under mild assumptions. The paper also derives the SBP in density regimes, obtaining a forward–backward Schrödinger system with and that determine the marginal dynamics, and provides explicit solvable cases (e.g., delta initial condition and Poisson) and a mollification-based approach for stable-like jump components. These results connect SBP with nonlocal operators, Lévy-type dynamics, and stochastic control, offering practical tools for inferring jump-diffusion bridges and enabling OT-inspired interpolation in jump-diffusion models.

Abstract

The Schrödinger bridge problem (SBP) seeks to find the measure on a certain path space which interpolates between state-space distributions at time and at time while minimizing the KL divergence (relative entropy) to a reference path measure . In this work, we tackle the SBP in the case when 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 -transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution as the strong-convergence limit of a sequence of harmonic -transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the -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.

Paper Structure

This paper contains 22 sections, 22 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Schrödinger Bridge Problem: Formulation and Motivation
  3. Assumptions (B)
  4. Main Results
  5. Connections between SBP and Other Fields
  6. Further Questions
  7. Static SBP and Diffusion SBP
  8. Static SBP and Schrödinger System
  9. Diffusion SBP and Dynamic Schrödinger System
  10. $h$-Transform of Jump Diffusions
  11. $h$-Transform: Preliminaries
  12. $h$-Transform: the Generator Approach
  13. $h$-Transform: the SDE Approach
  14. SBP for Jump Diffusions: the General Theory
  15. SBP with Explicit Solution
  16. ...and 7 more sections

Key Result

Theorem 2.2

follmer1988random The Schrödinger problems Definition:DynamicSBP and Definition:StaticSBP admit at most one solution $\hat{\mathbf{P}}$ and $\hat{\pi}$ respectively. Assume that the reference measure $R$ in the static problem Definition:StaticSBP and the reference path measure $\mathbf{R}$ in the dy

Theorems & Definitions (41)

  • Definition 1.1: Schrödinger Bridge Problem
  • Definition 2.1: Static Schrodinger Bridge Problem
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4: Fortet-Beurling-Jamison existence theorem
  • Theorem 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 31 more