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Adapted Wasserstein distance between the laws of SDEs

Julio Backhoff-Veraguas, Sigrid Källblad, Benjamin A. Robinson

TL;DR

The paper addresses how to quantify distance between laws of 1D SDEs while respecting information flow, by introducing the adapted Wasserstein distance $\\mathcal{A}\\mathcal{W}_p$ under bc-couplings. It proves the synchronous coupling is optimal under broad, 1D, pathwise-unique, and growth conditions and reveals a deep link to the Knothe--Rosenblatt rearrangement via a time-discretisation approach; it also establishes stability and discretisation convergence results, and shows that the induced topologies coincide on equicontinuous coefficient families. The work further provides numerical schemes that underpin the discrete-to-continuous connection and extends optimality to certain discontinuous drifts using Zvonkin’s transform. Through a suite of non-Markovian and multidimensional counterexamples, it clarifies the limitations of these optimality results and discusses alternative norms (e.g., $L^{\infty}$) where KR or synchronous optimality may fail, outlining the scope and practical impact for stochastic control, filtering, and finance where information flow is essential.

Abstract

We consider the bicausal optimal transport problem between the laws of scalar time-homogeneous stochastic differential equations, and we establish the optimality of the synchronous coupling between these laws. The proof of this result is based on time-discretisation and reveals a novel connection between the synchronous coupling and the celebrated discrete-time Knothe--Rosenblatt rearrangement. We also prove a result on equality of topologies restricted to a certain subset of laws of continuous-time processes. We complement our main results with examples showing how the optimal coupling may change in path-dependent and multidimensional settings.

Adapted Wasserstein distance between the laws of SDEs

TL;DR

The paper addresses how to quantify distance between laws of 1D SDEs while respecting information flow, by introducing the adapted Wasserstein distance under bc-couplings. It proves the synchronous coupling is optimal under broad, 1D, pathwise-unique, and growth conditions and reveals a deep link to the Knothe--Rosenblatt rearrangement via a time-discretisation approach; it also establishes stability and discretisation convergence results, and shows that the induced topologies coincide on equicontinuous coefficient families. The work further provides numerical schemes that underpin the discrete-to-continuous connection and extends optimality to certain discontinuous drifts using Zvonkin’s transform. Through a suite of non-Markovian and multidimensional counterexamples, it clarifies the limitations of these optimality results and discusses alternative norms (e.g., ) where KR or synchronous optimality may fail, outlining the scope and practical impact for stochastic control, filtering, and finance where information flow is essential.

Abstract

We consider the bicausal optimal transport problem between the laws of scalar time-homogeneous stochastic differential equations, and we establish the optimality of the synchronous coupling between these laws. The proof of this result is based on time-discretisation and reveals a novel connection between the synchronous coupling and the celebrated discrete-time Knothe--Rosenblatt rearrangement. We also prove a result on equality of topologies restricted to a certain subset of laws of continuous-time processes. We complement our main results with examples showing how the optimal coupling may change in path-dependent and multidimensional settings.
Paper Structure (17 sections, 22 theorems, 118 equations, 3 figures)

This paper contains 17 sections, 22 theorems, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary results on bicausal couplings and approximation in $\mathcal{A}\mathcal{W}_p$
  3. Characterisation of bicausal couplings between SDEs
  4. Stability of bicausal optimisers
  5. Approximation of SDEs in adapted Wasserstein distance
  6. The synchronous coupling: Properties and optimality
  7. The Knothe--Rosenblatt rearrangement
  8. A monotone numerical scheme
  9. $\mathcal{A}\mathcal{W}_p$ between laws of SDEs with continuous coefficients
  10. Extension to discontinuous drifts
  11. On the topology induced by the adapted Wasserstein distance
  12. Examples
  13. Non-Markovianity
  14. Multidimensional examples
  15. Adapted Wasserstein distance with $L^\infty$ norm
  16. ...and 2 more sections

Key Result

Theorem 1.3

Suppose that $(b, \sigma)$ and $(\bar{b}, \bar{\sigma})$ satisfy ass:main. Then, for any $p\ge 1$, the synchronous coupling attains the infimum in eq:adapted_wasserstein defining $\mathcal{A}\mathcal{W}_p(\mu^{b,\sigma},\mu^{\bar{b},\bar{\sigma}})$.

Figures (3)

  • Figure 1: Illustration of the Knothe--Rosenblatt rearrangement in two dimensions. The first marginals of $\mu$, $\nu$ are denoted $\mu_1$, $\nu_1$, and the conditional distributions by $\mu_{x_1}$, $\nu_{y_1}$. Similarly shaded regions have the same area.
  • Figure 2: The two possible trajectories of $X^n$, for some $n \in \mathbb N$, are shown on the left, and the two possible trajectories of $X^\infty$ on the right.
  • Figure 3: The Knothe--Rosenblatt rearrangement $\pi^\mathrm{KR}_{\mu, \nu}$ is shown on the left, and the coupling $\pi^\mathrm{AT}_{\mu, \nu}$ on the right. The solid blue lines and dashed orange lines represent processes with law $\mu$ and $\nu$, respectively. At each time, the points with the same colour and style are coupled with each other.

Theorems & Definitions (86)

  • Definition 1.1: bicausal couplings
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: correlated Wiener process
  • Proposition 2.2
  • proof
  • Remark 2.3: modified Wasserstein distance
  • Remark 2.4
  • Remark 2.5
  • ...and 76 more