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Bicausal optimal transport for SDEs with irregular coefficients

This paper establishes the optimality of synchronous coupling for bicausal optimal transport problems involving SDEs with irregular coefficients, providing a numerical method to compute adapted Wasserstein distances for model uncertainty quantification and introducing a new scheme with strong convergence guarantees for SDEs with exponentially growing and discontinuous drift.

Original authors: Michaela Hitz, Benjamin A. Robinson

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michaela Hitz, Benjamin A. Robinson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Comparing Two Different Worlds

Imagine you are a financial analyst or a scientist trying to predict the future. You have two different models (stories) about how a system behaves—let's say, the price of a stock or the spread of a virus.

Model A says the price moves smoothly.
Model B says the price jumps around wildly and sometimes changes its rules abruptly.

You want to know: How different are these two models?

In the past, mathematicians had a ruler to measure this difference called the Wasserstein distance. Think of this ruler as a way to measure how much "stuff" (probability) you have to move to turn one pile of sand into another.

The Problem: The old ruler is too simple. It only cares about the final shape of the piles. It doesn't care how the sand got there.

Analogy: Imagine two people walking from a house to a park.
- Person A walks straight there.
- Person B walks in circles, gets lost, and then arrives.
- If they both end up at the park at the same time, the old ruler says they are "close" because their starting and ending points are the same. But in reality, their journeys were totally different. In finance or biology, the journey (the path taken) matters just as much as the destination.

The Solution: The "Adapted" Ruler

The authors introduce a new, smarter ruler called the Adapted Wasserstein Distance.

The Rule: This ruler only allows you to compare two journeys if you respect the flow of time and information. You can't look at the future to decide the past.
The "Bicausal" Concept: Imagine two people walking side-by-side. A "bicausal" coupling means that at any specific moment, Person A's decision to turn left or right depends only on what they have seen so far, and the same for Person B. They can't cheat by peeking at the other person's future moves.

The Main Discovery: The "Synchronous" Strategy

The paper's biggest breakthrough is finding the perfect way to compare these two models.

Usually, calculating this "smart distance" is a nightmare. It's like trying to find the best way to rearrange a million pieces of a puzzle to minimize the total distance moved. It usually requires massive computer power and complex math.

The Authors' Finding:
For a huge class of models (specifically those described by Stochastic Differential Equations, or SDEs), the best way to compare them is surprisingly simple: Run them together.

The Metaphor: Imagine you have two runners (Model A and Model B). To see how different they are, you don't need to try every possible way to pair them up. You just need to make them run on the exact same track, at the exact same time, reacting to the exact same wind gusts.
This is called the Synchronous Coupling.
The authors prove that if you run these two models side-by-side using the same random noise (the "wind"), the distance between them is automatically the smallest possible "smart" distance.

Why This Matters: The "Irregular" Coefficients

Here is where the paper gets really tough. Most previous math could only handle "smooth" models (where the rules change gently). But the real world is messy.

Smooth: A car accelerating smoothly.
Irregular: A car that suddenly slams on the brakes, or a stock price that jumps because of a sudden news headline.

The authors tackle two types of "messy" math:

Discontinuous Drift: The rules change abruptly (like a step function).
Exponential Growth: The forces driving the system get huge very quickly (like a virus spreading or a bubble bursting).

The Challenge: Standard computer methods often crash or give wrong answers when dealing with these messy rules.

The Innovation:
The authors invented a new numerical scheme (a specific recipe for a computer to calculate the path).

They used a "transformation" trick. Imagine you have a bumpy, jagged road. Instead of trying to drive a car directly on the bumps, they first lay down a smooth, flat mat over the road (a mathematical transformation), drive the car on the mat, and then translate the result back to the real road.
They proved this new recipe works, converges to the right answer, and is stable even when the rules are broken or the numbers get huge.

The Practical Result: A New Tool for Uncertainty

Because they proved that "running them together" (Synchronous Coupling) is the best way to measure the difference, they created a fast, efficient method to calculate this distance.

What can you do with this?

Quantify Model Uncertainty: If you are unsure which model is correct (Model A or Model B), you can now calculate exactly how much your final decision might change if you picked the wrong one.
Optimal Stopping: This applies to problems like "When should I sell my stock?" or "When should I stop a clinical trial?" The paper shows that if your model of the world is slightly off, you can bound how much your "best time to stop" will change.

Summary in a Nutshell

The Problem: Comparing complex, time-based models (like stock prices) is hard because standard math ignores the "story" of how they got there.
The Fix: A new "Adapted" ruler that respects the flow of time.
The Secret: The best way to use this ruler is to simply run the two models side-by-side using the same random events.
The Breakthrough: They proved this works even when the models are "broken" or "wild" (discontinuous or exploding), and they built a computer recipe to do it accurately.
The Use: This helps experts measure how much their predictions might be wrong due to model uncertainty, specifically for things like financial options or biological processes.

Technical Summary: Bicausal Optimal Transport for SDEs with Irregular Coefficients

Problem Statement
The paper addresses the challenge of quantifying model uncertainty in stochastic systems described by Stochastic Differential Equations (SDEs). Specifically, it seeks to compute the distance between the laws of two SDEs, potentially possessing irregular coefficients (e.g., discontinuous or exponentially growing drift, non-Lipschitz diffusion). While the classical Wasserstein distance metrizes the weak topology of probability measures, it fails to capture the information structure (filtration) inherent in stochastic processes. This limitation renders it unsuitable for comparing models in contexts like mathematical finance or biology, where decisions depend on the flow of information. The authors focus on the adapted Wasserstein distance ( $AW_p$ ), which arises from a bicausal optimal transport problem. In this framework, couplings must respect the causal structure of the processes, ensuring that the future of one process is conditionally independent of the other given its past.

Methodology
The authors develop a methodology that bridges optimal transport theory and the numerical analysis of SDEs. The core strategy involves proving that the synchronous coupling—the joint law of two SDEs driven by a common Brownian motion—is the optimal transport plan for the adapted Wasserstein distance under specific, minimal regularity assumptions.

To establish this optimality for irregular coefficients, the paper employs two distinct transformation-based approaches corresponding to two classes of irregularity:

Discontinuous and Exponentially Growing Drift:
- Transformation: The authors utilize a spatial transformation $G$ (based on Leobacher and Szölgyenyi) to remove discontinuities in the drift coefficient. This maps the original SDE to a transformed SDE with continuous coefficients.
- Numerical Scheme: They introduce a transformation-based semi-implicit Euler–Maruyama scheme. This scheme is designed to be stochastically monotone, a crucial property for establishing optimality via the Knothe–Rosenblatt rearrangement.
- Convergence: The paper establishes the first strong convergence result for SDEs with drift that is both discontinuous and exponentially growing. The scheme is proven to converge strongly with known rates.
Bounded Measurable Drift and Hölder Continuous Diffusion:
- Transformation: The authors apply the Zvonkin transformation, which removes the drift term entirely, converting the SDE into one driven solely by a transformed diffusion coefficient.
- Optimality Transfer: They demonstrate that the Zvonkin transformation preserves the optimality property of the synchronous coupling. Since the transformed coefficients are sufficiently regular, existing optimality results apply, and the result is transferred back to the original SDE.

Key Contributions and Results

Optimality of Synchronous Coupling: The primary theoretical contribution is proving that the synchronous coupling attains the adapted Wasserstein distance for SDEs with:
- Discontinuous drift with exponential growth and degenerate diffusion (provided the diffusion is non-zero at discontinuity points).
- Bounded measurable drift and $\alpha$ -Hölder continuous diffusion ( $\alpha \in [1/2, 1]$ ).
- This generalizes previous results which were limited to globally Lipschitz or bounded coefficients.
Numerical Convergence:
- The paper provides a novel strong convergence result for a transformation-based semi-implicit Euler–Maruyama scheme applied to SDEs with discontinuous, exponentially growing drift.
- For $p \in [1, 2]$ , the scheme achieves a strong $L^p$ convergence rate of $1/2$ . For $p > 2$ , the rate is $1/(p(p-1))$ .
- This result fills a gap in the literature, as existing methods for such irregular coefficients often lacked uniform-in-time moment bounds or strong convergence guarantees.
Computational Efficiency:
- The authors show that computing the adapted Wasserstein distance reduces to simulating two one-dimensional SDEs driven by the same Brownian motion and applying Monte Carlo (or Multi-level Monte Carlo) methods.
- This approach avoids the computational intractability often associated with general optimal transport problems, where the optimal coupling is unknown.
Counterexamples and Limitations:
- The paper clarifies that the optimality of the synchronous coupling does not extend to all settings. Specifically, it provides a counterexample showing that for SDEs with non-Lipschitz diffusion (Hölder continuous), the monotone Euler–Maruyama scheme fails to be stochastically monotone.
- Furthermore, it demonstrates that replacing first-order stochastic dominance with second-order stochastic dominance in the definition of co-monotonicity is insufficient to guarantee the optimality of the Knothe–Rosenblatt rearrangement.

Significance and Applications
The paper claims that its results provide a computationally efficient method for quantifying model uncertainty in stochastic optimization problems. By establishing the optimality of the synchronous coupling under minimal regularity assumptions, the authors enable the practical calculation of the adapted Wasserstein distance for a broad class of SDEs that were previously intractable.

Specific applications highlighted include:

Robust Stochastic Optimization: The adapted Wasserstein distance serves as a metric for model uncertainty. The paper proves that optimal stopping problems are Lipschitz continuous with respect to this distance. This allows for the derivation of robust bounds on the value of optimal stopping problems (e.g., American-style Asian options) under different probability measures.
Numerical Examples: The authors validate their theoretical findings with numerical experiments on SDEs with discontinuous drift and the Cox–Ingersoll–Ross (CIR) model, demonstrating the convergence of the adapted Wasserstein distance to the perturbation size of the coefficients.

In summary, the work unifies optimal transport and numerical SDE analysis to solve a constrained transport problem, offering a rigorous and efficient tool for comparing stochastic models with irregular coefficients.