A PDE Derivation of the Schrödinger--Bass Bridge
Alexandre Alouadi, Pierre Henry-Labordère, Grégoire Loeper, Othmane Mazhar, Huyên Pham, Nizar Touzi
TL;DR
The paper addresses constructing optimal semimartingale couplings that interpolate between the Schrödinger bridge and the Bass martingale by introducing the Schrödinger--Bass Bridge (SBB) for $\beta>0$. It develops a direct PDE derivation in dimension one, using a sequence of convex-analytic transforms and the heat equation to obtain explicit representations of the time-marginals via a Schrödinger potential $h$ and a forward measure $\nu$, connected by monotone transport maps $\mathcal{Y}$ and $\mathcal{X}$. The main contributions are the SBB system, the interpretation of the transformed process $Y_t$ as a Schrödinger bridge and the stretched process $X_t = \mathcal{X}(t, Y_t)$, and a Sinkhorn-type algorithm for computing the solution; the framework recovers the Sinkhorn limit as $\beta\to\infty$ and the Bass limit as $\beta\to 0$. This provides a unified, PDE-driven lens for semimartingale optimal transport, with potential applications to diffusion calibration and related computational schemes. The results bridge classical optimal transport, Schrödinger bridges, and martingale transport, offering a tractable route to interpolate between these regimes while preserving explicit structural representations.
Abstract
This short paper announces the main results of \cite{SBB2026}, where the Schrödinger--Bass Bridge (SBB) problem is introduced and studied in full generality. Here we provide a direct PDE derivation of the SBB system in dimension one, showing how the optimal coupling problem that interpolates between the classical Schrödinger bridge and the Bass martingale transport can be solved explicitly via Legendre transforms and the heat equation. A key insight is that the optimal SBB process is a Stretched Schrödinger Bridge: the composition of a monotone transport map with a Schrödinger bridge. This extends the stretched Brownian motion representation of Bass martingales to the semimartingale setting and provides a unified framework that recovers both the Sinkhorn algorithm (in the limit $β\to \infty$) and the Bass construction (as $β\to 0$). We refer to \cite{SBB2026} for complete proofs, the multidimensional setting, strong duality, dual attainment, and further developments.