Adapted Wasserstein Distance

• Adapted Wasserstein distance is a refinement of classical optimal transport that restricts couplings via causality, preserving the temporal structure in stochastic processes.
• It transforms the space of filtered processes into a complete, geodesic metric space, enabling robust comparisons and displacement interpolations that respect time dynamics.
• Recent advances employ kernel smoothing and projection techniques to overcome computational challenges and achieve efficient empirical estimation in high-dimensional models.

The adapted Wasserstein distance is a refinement of the classical Wasserstein (optimal transport) distance that incorporates the temporal and informational structure of stochastic processes, making it particularly relevant for dynamic settings such as stochastic optimization, mathematical finance, and time-dependent machine learning tasks. By restricting the set of admissible couplings to those respecting a causality constraint—meaning only information available up to the current time may influence decisions—the adapted Wasserstein distance provides a more sensitive and robust measure of proximity between path laws and process-level distributions with respect to the flow of information and time.

1. Formal Definition and Distinction from Classical Wasserstein Distance

For stochastic processes or discrete-time path measures μ, ν on $(\mathbb{R}^d)^T$ (or more general Polish product spaces), the adapted p-Wasserstein distance, commonly denoted as $\mathcal{A}\mathcal{W}_p(\mu, \nu)$, is defined as

$$\mathcal{A}\mathcal{W}_p(\mu, \nu) = \left( \inf_{\pi \in \mathrm{Cpl}_{bc}(\mu, \nu)} \int \sum_{t=1}^T |x_t - y_t|^p \, d\pi(x, y) \right)^{1/p}$$

where $\mathrm{Cpl}_{bc}(\mu, \nu)$ is the set of bicausal couplings—joint measures $\pi$ on $((\mathbb{R}^d)^T)^2$ such that, for every $t$, the conditional law of $y_t$ given $x_{1:t}$ and $y_{1:t-1}$ depends only on past and present coordinates, and similarly for $x_t$ given $y_{1:t}$ and $x_{1:t-1}$ (Backhoff et al., 2020, Bartl et al., 2021).

This constraint enforces non-anticipativity: the transport plan cannot "look into the future," aligning the metric with dynamic programming and filtration-based frameworks encountered in stochastic control and mathematical finance. In contrast, the classical Wasserstein distance $\mathcal{W}_p$ minimizes over all joint couplings and hence may obscure dynamically relevant differences between processes that share similar marginals but diverge in temporal or information structure (Backhoff-Veraguas et al., 2019, Backhoff et al., 2020).

2. Structural Properties and Geometric Interpretation

The adapted Wasserstein distance turns the space $\mathrm{FP}_p$ of filtered processes with finite $p$-th moment into a complete and geodesic metric space (Bartl et al., 2021, Beiglböck et al., 28 Jun 2024). Geodesics—constant-speed interpolations between processes—are constructed using optimal bicausal couplings and are themselves filtered processes, allowing for displacement interpolation in path space that respects the temporal filtration. A notable consequence is that the set of martingale processes forms a closed, geodesically convex subset: the geodesic between any two martingales (under bicausal coupling) is a martingale (Bartl et al., 2021).

The adapted Wasserstein metric metrizes the so-called adapted weak topology, which is strictly finer than the weak topology induced by marginal distributions. This ensures continuity of key stochastic analysis operations (Doob decomposition, optimal stopping value functions, stochastic controls) that may be discontinuous in the classical topology (Bartl et al., 2021, Backhoff et al., 2020, Beiglböck et al., 28 Jun 2024).

Via isometry results, it is further shown that $(\mathrm{FP}_p, \mathcal{A}\mathcal{W}_p)$ is isomorphic to a classical Wasserstein space associated with the "information process," an object capturing the evolution of conditional laws over time (Bartl et al., 2021, Beiglböck et al., 28 Jun 2024).

3. Computation, Smoothing, and Empirical Approximation

Direct empirical estimation of the adapted Wasserstein distance is statistically and computationally challenging, primarily due to strong topological constraints—naive empirical measures generally do not converge in this metric (Backhoff et al., 2020, Hou, 26 Jan 2024). Recent advances have developed several remedying strategies:

• Smoothing via Kernel Convolution: Measures are first convolved with isotropic Gaussian noise to yield smoothed conditional kernels that are locally Lipschitz, overcoming the lack of regularity in high dimensions (Larsson et al., 13 Mar 2025, Hou, 26 Jan 2024). This smoothing leads to the smoothed adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(\sigma)}$, which achieves a fast, dimension-independent convergence rate $n^{-1/2}$ for subgaussian underlying measures, improving sharply on the $n^{-1/d}$ rate for classical Wasserstein in high dimensions (Larsson et al., 13 Mar 2025).
• Adapted Empirical and Smoothed Empirical Measures: Construction of empirical measures using smoothing and projection schemes (with data augmentation and translations to guarantee sufficient support) ensures convergence of these estimators in $\mathcal{A}\mathcal{W}_p$ with explicit deviation bounds (Hou, 26 Jan 2024). The combination of kernel smoothing, adapted projection, and random shifting is necessary for both statistical convergence and practical estimation.
• Reduction to Classical Transport via Regularity: For measures with smooth (Sobolev) densities, $\mathcal{A}\mathcal{W}_p$ can be controlled by a bi-Lipschitz estimate in terms of the classical Wasserstein distance, especially after smoothing. Under suitable regularity, the adapted total variation distance is comparable to the classical total variation distance with constants linear in the time horizon (Acciaio et al., 27 Jun 2025, Blanchet et al., 31 Jul 2024).

4. Theoretical Guarantees, Bounds, and Interpolation Properties

Several quantitative bounds have been established connecting the adapted and classical Wasserstein distances:

• Explicit Upper Bounds: The adapted Wasserstein distance is bounded from above by a function of the classical Wasserstein distance, the regularity modulus (e.g., Lipschitz constant) of conditional kernels, and tail behavior (Blanchet et al., 31 Jul 2024). For measures with Lipschitz kernels, $$\mathcal{A}\mathcal{W}_1(\mu, \nu) \leq C \sqrt{\mathcal{W}_1(\mu, \nu)}$$, where $C$ depends on structural and regularity constants (Blanchet et al., 31 Jul 2024).
• Smooth Adapted Wasserstein and Topology Interpolation: The smoothed adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(\sigma)}$ defines a topology interpolating between the classical Wasserstein and the adapted Wasserstein topology. For fixed noise $\sigma > 0$, it is equivalent to the classical topology; as $\sigma \to 0$, it converges to the adapted topology, except at a critical rate depending on the regularity of the conditional kernels (Blanchet et al., 31 Jul 2024, Larsson et al., 13 Mar 2025).
• Transport-Entropy (T₁) Inequality: The adapted T₁ inequality provides a concentration of measure result for bicausal couplings: for a process law $\mu$ satisfying an exponential moment condition, one has $$\mathcal{A}\mathcal{W}_1(\mu, \nu) \leq C \sqrt{2\, H(\nu|\mu)}$$ for all $\nu$, with $C$ scaling as $2\sqrt{T} + 1$ in the number of time steps $T$ (logarithmic factors omitted), mirroring the classical Bolley–Villani inequality but refined to bicausal transport (Park, 25 Jul 2025).

5. Closed-Form Formulas and Specialized Examples

Closed-form expressions for the adapted 2-Wasserstein distance between Gaussian process laws have been established, highlighting the structural divergence from the classical setting:

• For $\mu = \mathcal{N}(a, A)$ and $\nu = \mathcal{N}(b, B)$ on $\mathbb{R}^N$, the adapted squared 2-Wasserstein distance is

$$\mathcal{A}\mathcal{W}_2^2(\mu, \nu) = \|a - b\|^2 + \mathrm{Tr}(A + B) - 2\|\mathrm{diag}(L^T M)\|_1$$

where $A = L L^T$, $B = M M^T$ are Cholesky factors and $\|\cdot\|_1$ is the $\ell_1$-norm of the diagonal. The optimizing coupling is highly structured, reflecting sequential dependencies (Gunasingam et al., 9 Apr 2024, Acciaio et al., 25 Dec 2024, Jiang et al., 27 May 2025).

• Entropic regularization can be included, yielding the entropic adapted Wasserstein distance $AW_{2,\lambda}$ which admits a closed-form formula for multidimensional Gaussian processes, involving explicit expressions using singular values and block-diagonal operator functions (Acciaio et al., 25 Dec 2024).
• In infinite dimensions, the adapted transport between Gaussian processes may be characterized via causal operator factorizations that generalize the finite-dimensional Cholesky decomposition, enabling computations for mean-square continuous Volterra or even fractional Brownian motion processes (Jiang et al., 27 May 2025).

6. Applications in Optimization, Finance, and Learning

• Stochastic Optimization: The adapted Wasserstein distance controls the sensitivity of the value functions in multi-period and optimal stopping problems. Under model uncertainty, it quantifies the maximal shift in the optimized value as the model law is perturbed within an adapted Wasserstein ball, with explicit first-order approximations (risk measures) available (Bartl et al., 2022).
• Financial Mathematics: In robust pricing and hedging, the adapted Wasserstein distance determines the stability of superhedging strategies and utility maximization with respect to model perturbations. Unlike the classical metric, adapted Wasserstein closeness implies financial (not only statistical) similarity by preserving the temporal dynamics and filtration (Backhoff-Veraguas et al., 2019).
• Statistical Testing: Empirical martingale projection distances calculated via the adapted Wasserstein distance allow for consistent and efficient hypothesis testing of the martingale property (e.g., validating no-arbitrage in neural SDE-based asset pricing), even in high dimensions (Blanchet et al., 22 Jan 2024).
• Machine Learning: In sequential learning and domain adaptation, adapted Wasserstein distances can be used for distribution comparison and robust representation learning to incorporate dynamic and causal information. Smoothing and embedding strategies facilitate scalable computation and make the metric usable in practice (Shen et al., 2017, Courty et al., 2017).

7. Extensions, Open Directions, and Limitations

Research continues to deepen the connection between adapted and classical optimal transport, develop efficient empirical estimators, and transfer statistical guarantees from classical to adapted settings (Blanchet et al., 31 Jul 2024, Acciaio et al., 27 Jun 2025). Limitations include computational cost for non-Gaussian or high-dimensional path measures, and the topological and statistical complexity of estimation—even for seemingly simple stochastic processes, empirical measures may not converge except under suitable adaptations (smoothing, shifting, regularization) (Backhoff et al., 2020, Hou, 26 Jan 2024, Larsson et al., 13 Mar 2025).

The adapted Wasserstein distance is thus a core object in modern probability and optimization in the presence of time/filtration structure, providing both geometric insight and robust quantitative tools for dynamic problems across mathematics, statistics, and engineering.