[Paper Review] A PDE Derivation of the Schrödinger--Bass Bridge

Alexandre Alouadi, Pierre Henry-Labordère+4 more|arXiv (Cornell University)|Jan 25, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper provides a direct PDE derivation in 1D showing that the Schrödinger–Bass Bridge (SBB) is solved via Legendre transforms and the heat equation, revealing a Stretched Schrödinger Bridge representation that interpolates between Sinkhorn and Bass constructions.

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 $β o \infty$) and the Bass construction (as $β o 0$). We refer to \cite{SBB2026} for complete proofs, the multidimensional setting, strong duality, dual attainment, and further developments.

Motivation & Objective

• Motivate the Schrödinger–Bass Bridge (SBB) as an interpolation between Schrödinger bridge and Bass martingale problems.
• Derive the SBB system in dimension one using a PDE approach and Legendre duality.
• Show that the optimal SBB process can be represented via a Stretched Schrödinger Bridge.
• Provide a unified framework that recovers Sinkhorn in the large-β limit and Bass in the small-β limit.

Proposed method

• Formulate the primal SBB problem with drift and volatility penalties and derive the dual HJB equation.
• Perform a sequence of transformations: define u from v, apply the Legendre transform to obtain u*, and reduce to the heat equation via w and h.
• Introduce the transport maps 𝒴 and 𝒳 as gradients of convex potentials tied to h, yielding explicit expressions for the optimal drift and diffusion.
• Show that Y_t follows a Schrödinger bridge with potential h solving the backward heat equation, and X_t is obtained by applying a monotone transport to Y_t.
• Present the SBB system that couples the time-marginals μ_t and the Schrödinger component h via the map 𝒳 and the forward/backward heat equations.
• Outline an iterative Sinkhorn-type scheme to enforce marginals using the Monge–Ampère structure derived in the setting.

Experimental results

Research questions

• RQ1 How can the Schrödinger–Bass Bridge be derived directly from PDEs in one dimension?
• RQ2 How are Legendre duality and the heat equation used to obtain explicit representations of the SBB system?
• RQ3 Can the SBB be interpreted as a Stretched Schrödinger Bridge that unifies Schrödinger bridge and Bass martingale approaches?
• RQ4 What are the limiting behaviors of the SBB with β → ∞ and β → 0, and how do they recover Sinkhorn and Bass constructions respectively?
• RQ5 How can the SBB solution be computed numerically via a Sinkhorn-type algorithm?

Key findings

• The SBB solution in 1D can be represented as a Stretched Schrödinger Bridge, where Y_t is a Schrödinger bridge and X_t is obtained by a monotone transport of Y_t.
• The optimal dynamics are given by drift α(t,x)=∂_x v(t,x)=β(x−𝒴(t,x)) and diffusion σ(t,x)=1/(1−∂_{xx}v/β), linking to the maps 𝒴 and 𝒳.
• A sequence of transformations (u, u*, w, h) reduces the HJB structure to the linear heat equation for h, enabling explicit construction.
• The time-marginals satisfy μ_t = 𝒳(t,.)_# (h(t,.) ν_t) with h solving ∂_t h + 1/2 ∂_{yy} h = 0 and ν solving ∂_t ν = 1/2 ∂_{xx} ν.
• The SBB system unifies Sinkhorn and Bass, recovering Schrödinger in the large-β limit and Bass in the small-β limit via the behavior of 𝒳 and h.
• An iterative Sinkhorn-type scheme is proposed for computing the SBB solution by alternating maps and density updates.