Analytical Approach to Continuous-Time Causal Optimal Transport

Abstract: We study causal optimal transport in continuous time, with Markovian cost, between a finite-state Markov source and a diffusion target. By replacing the source with its conditional law given the observation of the target, we characterize the value of this transport problem through a fully nonlinear parabolic master equation on an enlarged state space. We further show that this value coincides with those of two equivalent stochastic control problems on the simplex: a control of the Kushner--Stratonovich filtering equation with a zero-mean condition, and a state-constrained stochastic optimal control problem. Both formulations give rise to implementable numerical schemes that approximate the value from above and below.

• The paper establishes the equivalence between the causal OT problem, state-constrained control, and filtering formulations using a novel filter representation.
• It reformulates the problem into a fully nonlinear parabolic PDE whose viscosity solution yields an explicit value function with rigorous approximation schemes.
• Robust numerical and duality frameworks are provided, ensuring monotone convergence and practical computation through discretized tree-based schemes.

Analytical Characterization of Continuous-Time Causal Optimal Transport

Motivation and Context

Optimal transport (OT) theory, classically formulated for measures and static probability distributions, has seen growing application to stochastic processes in fields ranging from mathematical finance to machine learning. However, traditional OT lacks sensitivity to the temporal directionality and filtration structure intrinsic to stochastic processes. Wasserstein distances, for instance, may fail to distinguish between processes with disparate underlying adaptiveness and information flow, leading to mismatches in applications to optimal stopping, stochastic control, or other time-sensitive decision problems.

To rectify this, causal OT imposes adaptation constraints on couplings: the conditional law of the target process given the source must respect the time filtration, ensuring that future increments of the target depend only on past and current states of the source. This framework is nontrivial in continuous time, where the theoretical and practical characterization of the associated values and optimizers are considerably underdeveloped compared to discrete time. The present paper addresses this lacuna.

Problem Statement and Reformulation

The causal OT problem investigated here is defined between a finite-state continuous-time Markov chain $X$ (source) and a diffusion process $Y$ (target) on $\mathbb{R}^d$. The cost functional is Markovian and integrates both running and terminal costs. Admissible couplings $\pi$ must preserve the source and target marginals, with causality enforced via filtration-based measurability: for every $t$, the conditional law of $Y$ given $X$ up to time $t$ is measurable with respect to the history of $X$.

A critical asymmetry emerges: any causal coupling preserves the Markovian semimartingale structure of $X$ within the joint filtration, but $Y$0 may acquire additional drift under the enlarged filtration unless further constraints are imposed. The correct state enlargement is identified as the filter process $Y$1, the conditional law of $Y$2 given $Y$3, evolving stochastically in the simplex $Y$4.

By reformulating the original OT problem in terms of this filter, and expressing the objective as a linear functional of the pair $Y$5, the authors establish an equivalence between the causal OT problem and two stochastic control formulations:

• A state-constrained control problem, where $Y$6 evolves via a generalized filtering/innovation equation under admissible controls.
• A control of the Kushner–Stratonovich filtering equation, subject to a zero-mean constraint on the innovation (reflecting fixed law of $Y$7).

Both formulations yield the same value as the original causal OT problem, with the filter-based state space providing a tractable representation.

Main Results and PDE Characterization

The principal outcome is a rigorous analytical characterization of continuous-time causal OT via a fully nonlinear parabolic PDE on the enlarged state space $Y$8. The unique value function is shown to be the viscosity solution of a variational inequality involving:

• A Hamilton–Jacobi–Bellman operator reflecting state-constrained optimal control,
• Boundary conditions encoding the simplex constraint inherent in the filter process,
• Explicit appearance of the zero-conditional-mean condition on controls, ensuring the law of $Y$9 remains prescribed.

The main theorem confirms the coincidence of the value for the original causal OT problem, the filtering lower bound, and the state-constrained upper bound, and provides constructive PDE-based numerical schemes for computing the value. Strong uniqueness is established via a comparison principle for the viscosity solutions.

Approximation and Duality

The PDE representation enables both monotone approximation schemes (from above and below) for practical computation:

• Upper bounds: via dual representations and truncating controls, yielding sequences of PDEs with bounded controls converging to the true value.
• Lower bounds: via limiting arguments on the state-constrained control problem, allowing for discretized numerical implementations.

A complementary duality framework is provided, reducing the problem to optimization over terminal filters and elucidating connections to martingale transport and free-boundary stochastic control problems.

Examples and Numerical Results

Explicit examples demonstrate the analytical tractability and numerical computation potential of the approach. Special cases including constant-state Markov chains, absorbing states, and irreducible two-state chains are treated in detail, linking the optimal value to classical rearrangement and monotone follower problems. Numerical discretization via tree-based schemes provides validation for the monotone approximation results, exhibiting rapid convergence and explicit sandwiching of the true value between primal and dual schemes.

Theoretical Implications

The identification of the filter as the correct geometric state variable, and the reduction to a finite-dimensional stochastic control problem on the simplex, resolve the longstanding complexity of the causal OT constraint in continuous time. The PDE characterization bridges gaps between causal transport, stochastic filtering, stochastic control, and information theory, incorporating intricate constraints arising from fixed marginals and zero-mean innovation.

Furthermore, the framework subsumes and generalizes previous results in discrete time and bicausal settings. The comparison with bicausal OT reveals deep structural differences: causal OT requires state space enlargement, while bicausal OT's symmetry admits direct PDE characterizations. The presence of the zero-conditional-mean constraint is structurally analogous to the inconspicuous trading condition in asymmetric information models.

Practical Implications and Extensions

The analytic and numerical schemes developed provide explicit routes for statistical estimation and stability assessment in applications such as time-series analysis, finance, and causal inference for sequential models. The ability to compute and approximate causal OT distances in continuous time opens the door for robust optimization methods, statistical consistency theorems, and stability analysis in high-dimensional settings.

Extensions to non-Markovian sources, continuous state spaces, and path-dependent costs are outlined. Such generalizations, although technically challenging due to infinite-dimensional state constraints and the geometry of path-space measures, have profound implications for process-based adapted OT distances and information-theoretic regularization in stochastic modeling.

Conclusion

This paper presents a comprehensive and rigorous analytical framework for continuous-time causal optimal transport between finite-state Markov sources and diffusion targets. By delineating the filter-based enlargement of state space, and establishing equivalence with state-constrained and filtering-based control problems, the authors enable the first explicit PDE characterization and practical computation of causal OT values in continuous time. Theoretical and numerical approximation results provide immediate utility for applications in stochastic analysis and related domains, with extensions poised to advance the field of adaptive transport and process-based probability metrics.

---

Strong numerical results include the monotone convergence of both upper and lower approximation sequences to the PDE-defined value and the practical implementation via discretized tree-based schemes, sandwiching the true value within quantifiable bounds.

Bold claims in the paper: The value of the continuous-time causal OT problem coincides exactly with both the filtering lower bound and the state-constrained upper bound; causal OT can be reduced to a finite-dimensional stochastic control problem on the simplex; the solution to the associated PDE yields the explicit value function, uniquely characterized by viscosity solution theory.

Future developments: Potential extensions include infinite-dimensional process state spaces (requiring novel comparison principles and PDE techniques) and non-Markovian filtrations, paving the way for causal OT between general path measures. Adapted Wasserstein distances and associated computations stand to benefit from the analytical machinery developed herein.