Abstract

We study the problem of generating a continuous semimartingale that interpolates prescribed initial and terminal distributions. We cast this task as an optimal transport problem on path space that bridges the Schrödinger bridge and the Bass martingale formulations, yielding a diffusion whose drift and volatility are jointly calibrated to data. We derive an analytic characterization of the optimizer—Schrödinger–Bridge–Bass (SBB) diffusion—and show that it admits a representation as a stretched Brownian motion under an explicit change of measure. Building on this structure, we develop a practical computational scheme for SBB that is efficient and scalable, providing a principled framework for generative diffusion modeling. We outline applications to image synthesis and time‑series generation, highlighting SBB as a unifying lens connecting entropic and martingale optimal transport in modern generative modeling.