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Bicausal optimal transport for SDEs with irregular coefficients

Michaela Hitz, Benjamin A. Robinson

TL;DR

The paper develops a framework to compare laws of SDEs with irregular coefficients using the adapted Wasserstein distance, showing that the synchronous coupling is optimal among bicausal couplings in several irregular settings. It introduces a transformation-based, strongly convergent semi-implicit Euler–Maruyama scheme to handle discontinuous and exponentially growing drifts, enabling efficient computation of the adapted Wasserstein distance via common-noise simulations and Monte Carlo methods. The results cover two main irregularity classes (growth-disc with discontinuous drift and Zvonkin regularisation for bounded drift) and establish general optimality principles that support robust optimization applications. Practically, the work provides both theoretical guarantees and numerical tools for model uncertainty quantification in stochastic optimization and robust stopping problems, with precise convergence and stability results tied to the transformed SDEs and their discretisations.

Abstract

We solve constrained optimal transport problems in which the marginal laws are given by the laws of solutions of stochastic differential equations (SDEs). We consider SDEs with irregular coefficients, making only minimal regularity assumptions. We show that the so-called synchronous coupling is optimal among bicausal couplings, that is couplings that respect the flow of information encoded in the stochastic processes. Our results provide a method to numerically compute the adapted Wasserstein distance between laws of SDEs with irregular coefficients. We show that this can be applied to quantifying model uncertainty in stochastic optimisation problems. Moreover, we introduce a transformation-based semi-implicit numerical scheme and establish the first strong convergence result for SDEs with exponentially growing and discontinuous drift.

Bicausal optimal transport for SDEs with irregular coefficients

TL;DR

The paper develops a framework to compare laws of SDEs with irregular coefficients using the adapted Wasserstein distance, showing that the synchronous coupling is optimal among bicausal couplings in several irregular settings. It introduces a transformation-based, strongly convergent semi-implicit Euler–Maruyama scheme to handle discontinuous and exponentially growing drifts, enabling efficient computation of the adapted Wasserstein distance via common-noise simulations and Monte Carlo methods. The results cover two main irregularity classes (growth-disc with discontinuous drift and Zvonkin regularisation for bounded drift) and establish general optimality principles that support robust optimization applications. Practically, the work provides both theoretical guarantees and numerical tools for model uncertainty quantification in stochastic optimization and robust stopping problems, with precise convergence and stability results tied to the transformed SDEs and their discretisations.

Abstract

We solve constrained optimal transport problems in which the marginal laws are given by the laws of solutions of stochastic differential equations (SDEs). We consider SDEs with irregular coefficients, making only minimal regularity assumptions. We show that the so-called synchronous coupling is optimal among bicausal couplings, that is couplings that respect the flow of information encoded in the stochastic processes. Our results provide a method to numerically compute the adapted Wasserstein distance between laws of SDEs with irregular coefficients. We show that this can be applied to quantifying model uncertainty in stochastic optimisation problems. Moreover, we introduce a transformation-based semi-implicit numerical scheme and establish the first strong convergence result for SDEs with exponentially growing and discontinuous drift.
Paper Structure (26 sections, 26 theorems, 146 equations, 5 figures)

This paper contains 26 sections, 26 theorems, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Optimal transport and applications to stochastic processes
  4. Introduction to optimal transport
  5. Brenier's theorem
  6. Wasserstein distance
  7. Optimal transport applied to stochastic processes
  8. Bicausal optimal transport
  9. Optimality of bicausal couplings
  10. Continuous-time bicausal optimal transport
  11. Numerical methods for SDEs
  12. Preliminary results
  13. Discontinuous drift with exponential growth
  14. A transformed SDE
  15. Transformed semi-implicit Euler--Maruyama scheme
  16. ...and 11 more sections

Key Result

Proposition 2.8

A coupling $\pi \in \mathrm{Cpl}(\mu, \nu)$ is equal to $\pi^\mathrm{KR}_{\mu, \nu}$ if and only if $\pi = \mathrm{Law}(X, Y)$, where for $U_1$, $\dotsc\,$, $U_n$ independent uniform random variables on $[0, 1]$, and functions $T_i, S_i \colon \mathbb R^i \to \mathbb R$ that are co-monotone in $U_i$, for each $i \in \{1, \dotsc, n\}$.

Figures (5)

  • Figure 1: A sequence of two-step deterministic processes $X^\varepsilon$ converges weakly as $\varepsilon \to 0$ to a martingale $X$BaBaBeEd19a.
  • Figure 2: From left to right: the functions $G$, $G-\operatorname{id}$, $G'$, and $G"$, for $\xi = 0$, $\alpha = 1$, $c = 1/6$.
  • Figure 3: Two couplings of the laws of $X$ and $\bar{X}$. In each case, paths with the same colour and line style are coupled with each other.
  • Figure 4: Approximation of the adapted Wasserstein distance for the SDEs \ref{['eq:disc']}.
  • Figure 5: Approximation of the adapted Wasserstein distance for the CIR process; the dotted purple line shows the effect of a perturbed diffusion parameter $\gamma$, the solid blue line shows the effect of a perturbed mean reversion level $\eta$, and the dashed green line shows the effect of a perturbed mean reversion speed $\kappa$.

Theorems & Definitions (79)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3: adapted Wasserstein distance -- discrete time
  • Remark 2.4
  • Example 2.5: \ref{['ex:usual']} revisited
  • Definition 2.6: stochastic co-monotonicity Da68
  • Definition 2.7: Knothe--Rosenblatt rearrangement
  • Proposition 2.8
  • Proposition 2.9
  • Remark 2.10
  • ...and 69 more