Stretched Schrödinger Bridge

• Stretched Schrödinger Bridge is a semimartingale process that optimally interpolates between classical entropic interpolation and Bass martingale transport by composing a monotone transport map with a Schrödinger bridge.
• It is defined via a mixed-penalty variational formulation where the parameter β controls the trade-off between enforcing constant volatility and optimizing drift under martingale constraints.
• Coupled nonlinear PDEs and Legendre transform techniques linearize the problem, thereby enabling explicit constructions, numerical Sinkhorn-type algorithms, and applications in stochastic control and financial calibration.

The Stretched Schrödinger Bridge is a semimartingale process that optimally interpolates between the classical Schrödinger bridge (entropic interpolation) and the Bass (stretched Brownian) martingale transport. This process is constructed as the composition of a monotone transport map with a Schrödinger bridge, yielding a solution that unifies entropic and martingale optimal transport in a single parametric framework. The one-dimensional Stretched Schrödinger–Bass Bridge (SBB) arises as the optimizer of a mixed-penalty variational problem and admits an explicit PDE characterization via Legendre transforms and the heat equation, revealing deep connections between stochastic control, convex analysis, and semimartingale optimal transport (Alouadi et al., 25 Jan 2026).

1. Variational Formulation of the SBB Problem

Given two probability measures $\mu_0$ and $\mu_T$ on $\mathbb{R}$ with finite second moments and a parameter $\beta>0$, the SBB problem seeks a continuous semimartingale $X_t$ ($0\leq t \leq T$) with dynamics:

$$dX_t = \alpha_t\,dt + \sigma_t\,dW_t, \qquad X_0\sim\mu_0,\ X_T\sim\mu_T,$$

minimizing the mixed-penalty cost functional

$$\mathrm{SBB}(\mu_0,\mu_T) = \inf_{(\alpha,\sigma)}\,\mathbb{E}\left[\frac{1}{2}\int_0^T (|\alpha_t|^2 + \beta|\sigma_t-1|^2)dt\right].$$

The parameter $\beta$ determines the SBB regime:

• As $\beta\to\infty$, the volatility penalty enforces $\sigma_t\approx 1$, recovering the classical Schrödinger bridge (entropic interpolation).
• As $\beta\to 0$, the volatility is unconstrained while the drift optimizes martingale constraints, yielding the Bass (stretched Brownian) martingale transport.

2. Coupled Partial Differential Equation System

The dual formulation introduces a Hamilton–Jacobi–Bellman (HJB) value function $v(t,x)$ satisfying the nonlinear PDE:

$$\partial_t v + \frac{1}{2}|\partial_x v|^2 + \frac{1}{2}\,\frac{\partial_{xx}v}{1-(\partial_{xx}v)/\beta} = 0, \quad \partial_{xx}v < \beta.$$

A change of variables $u(t,x) := x^2/2 - v(t,x)/\beta$ yields a convex function $u$ solving

$$\partial_t u - \frac{\beta}{2}|\partial_x u - x|^2 + \frac{1}{2}(1 - 1/\partial_{xx}u) = 0.$$

The Legendre transform $u^*(t,y)$ in $x$,

$u^*(t,y) := \sup_{x\in\mathbb{R}}\,[xy - u(t,x)],$

leads to the key transformation $h(t,y) := \exp(\beta(u^*(t,y) - y^2/2))$, which satisfies the linear backward heat equation:

$\partial_t h + \frac{1}{2}\partial_{yy} h = 0.$

Simultaneously, the forward density $\nu(t,y)$ of the base Brownian motion satisfies the forward heat equation:

$\partial_t \nu - \frac{1}{2}\partial_{yy} \nu = 0,$

with boundary pushforward constraints ensuring compatibility with $\mu_0$ and $\mu_T$.

3. Explicit Construction: Monotone Transport and Schrödinger Bridge

The optimal SBB is realized by coupling a classical Schrödinger bridge with a monotone, time-dependent transport map. Define

$$\begin{aligned} \mathcal{Y}(t,x) &:= \arg\min_{y\in\mathbb{R}} \bigl[\log h(t,y) + \frac{\beta}{2}|x-y|^2\bigr],\ \mathcal{X}(t,y) &:= y + (1/\beta)\partial_y\log h(t,y). \end{aligned}$$

First-order optimality conditions ensure the invertibility relations $\mathcal{X}(t,\mathcal{Y}(t,x))=x$, $\mathcal{Y}(t,\mathcal{X}(t,y))=y$.

The backward potential $h(t,y)$ and forward density $\nu(t,y)$ play the role of Schrödinger potentials, solving the usual forward–backward heat system. If $Y_t$ is the Schrödinger bridge with dynamics

$dY_t = \partial_y\log h(t,Y_t)\,dt + dW_t,$

the optimal SBB process is given by

$X_t = \mathcal{X}(t, Y_t).$

This establishes the Stretched Schrödinger Bridge as the composition of the base Schrödinger bridge $Y_t$ with the monotone map $\mathcal{X}$ (Alouadi et al., 25 Jan 2026).

4. Limiting Regimes and the Role of $\beta$

The parameter $\beta$ governs the interpolation between entropic and martingale couplings:

• As $\beta\to\infty$: $\mathcal{X}(t,y)\to y$, so $X_t\to Y_t$. The SBB reduces to the Schrödinger bridge and recovers Sinkhorn-type scaling.
• As $\beta\to 0$: the backward heat potential $h$ becomes constant and $Y_t$ becomes standard Brownian motion; $\mathcal{X}(t,y)$ becomes a convex map corresponding to the Bass (stretched Brownian) martingale transport.

This parameterization creates a continuum of interpolating couplings between purely entropic and purely martingale scenarios, controlled by the mixed-penalty cost in the SBB problem.

5. Duality, Strong Duality, and Heat Equation Linearization

The SBB admits both primal and dual variational formulations:

• Primal (P): Minimize over all admissible $(\alpha, \sigma)$ with law endpoints,

$$\mathrm{SBB}(\mu_0,\mu_T) = \inf_{(\alpha,\sigma)} \mathbb{E}[\frac{1}{2}\int_0^T (|\alpha_t|^2 + \beta|\sigma_t-1|^2)dt],\quad X_0\sim\mu_0,\ X_T\sim\mu_T.$$

• Dual (D): Supremum over sufficiently regular $v$ solving the HJB PDE,

$$V(\mu_0,\mu_T) = \sup_{v} \left\{ \mathbb{E}_{\mu_T}[v(T,X)] - \mathbb{E}_{\mu_0}[v(0,X)] \right\}, \quad v \text{ solves HJB, } \partial_{xx}v < \beta.$$

Strong duality holds ($\mathrm{SBB} = V$), and existence of an optimal control is established. A notable feature is the reduction of the nonlinear HJB system, via convex and Legendre transforms, to coupled linear heat equations for $h$ and $\nu$. The classical SDE and transformation structure are explicitly characterized:

$$X_t = \mathcal{X}(t, Y_t), \qquad \mathcal{X}(t,y) = \partial_y \left( \frac{y^2}{2} + \frac{1}{\beta} \log h(t,y) \right).$$

6. Theoretical and Practical Implications

The SBB framework offers a unified approach to semimartingale optimal transport, interpolating smoothly between entropic and martingale couplings as $\beta$ varies. In the one-dimensional case, the SBB enables linearization of the nonlinear HJB–Fokker–Planck system into decoupled heat equations supplemented by evolving Monge maps, providing tractable analytic and computational solutions. This synthesis extends the stretched Brownian realization of Bass martingales and the classical Schrödinger bridge, opening pathways to numerical Sinkhorn-type algorithms, financial model calibration, and potential multidimensional and multiperiod generalizations. The explicit connection to monotone transport and the heat equation establishes new structural and algorithmic possibilities in semimartingale optimal transport (Alouadi et al., 25 Jan 2026).