Yamada-Watanabe-Engelbert Theorem

• The theorem establishes conditions where weak existence combined with pathwise uniqueness guarantees strong existence and uniqueness in law.
• It employs compatibility structures and measurable selection techniques to extend classical one-dimensional SDE results to multidimensional, jump, and infinite-dimensional models.
• Applications include Itô diffusions, particle systems, and financial models, unifying diverse stochastic frameworks under one abstract paradigm.

The Yamada–Watanabe–Engelbert theorem provides a rigorous framework for characterizing the relationships between weak and strong solutions of stochastic (and, more generally, random input–output) equations, establishing precise conditions under which (i) pathwise uniqueness and weak existence imply strong existence and uniqueness in law, and (ii) under certain duality structures, strong existence and uniqueness in joint law imply pathwise uniqueness. This theorem unifies and generalizes the classical results of Yamada–Watanabe for one-dimensional SDEs, Engelbert–Schmidt for weak existence, and a broad class of infinite-dimensional, jump, time-change, and particle system models via the abstract “compatibility structure” paradigm introduced by Kurtz. Modern versions extend to SPDEs in Banach or Hilbert spaces and systems with irregular coefficients.

1. Foundational Definitions and General Abstract Framework

The abstract Yamada–Watanabe–Engelbert paradigm begins with a general stochastic model relating random “inputs” $Y$ (with law $\nu$ on Polish space $S_2$) to “outputs” $X$ (in space $S_1$), subject to a collection $\Gamma$ of joint constraints (e.g., integral, differential, or pathwise criteria) that define the notion of solution. Weak solutions are defined as pairs $(X,Y)$ on some probability space with $\mathrm{Law}(Y) = \nu$ and joint law in $S_{\Gamma,\nu}$. Strong solutions additionally require the existence of a Borel map $F\:S_2\to S_1$ such that $X = F(Y)$ almost surely (Kurtz, 2013).

A key technical innovation is the introduction of compatibility structures $$C = \{(\mathcal{B}_\alpha^{S_1},\mathcal{B}_\alpha^{S_2})\}$$, which generalize classical adaptedness or independence requirements: $X$ is $C$-compatible with $Y$ if, for every $\alpha$ and bounded measurable $h: S_2 \to \mathbb{R}$,

$$\mathbb{E}[h(Y) | \mathcal{G}_\alpha^X \vee \mathcal{G}_\alpha^Y] = \mathbb{E}[h(Y) | \mathcal{G}_\alpha^Y].$$

This encodes temporal filtration, martingale preservation, or more exotic relations, allowing the framework to encompass Itô diffusions, semimartingale-driven SDEs, McKean–Vlasov equations, and time-changed stochastic systems (Kurtz, 2013).

2. Main Theorems: Equivalence of Weak Pathwise Uniqueness and Strong Existence

The abstract theorem, as formalized by Kurtz, asserts:

Theorem (General Yamada–Watanabe–Engelbert). Let $S_{\Gamma, C, \nu}$ denote the set of $C$-compatible joint laws with $Y$-marginal $\nu$ and satisfying constraints $\Gamma$. Then, the following are equivalent:

• (a) $S_{\Gamma,C,\nu}\ne\emptyset$ (i.e., a weak solution exists) and pointwise uniqueness: any two $C$-compatible solutions with same input coincide almost surely.
• (b) There exists a strong solution $X = F(Y)$ a.s., and joint uniqueness in law (only one such joint law).

Consequences include the existence of a measurable selector $F$ so that any admissible solution must satisfy $X = F(Y)$ a.s., and, conversely, that under strong solutions with joint uniqueness, pathwise uniqueness follows (Kurtz, 2013).

For classical SDEs, this reduces to “weak existence + pathwise uniqueness $\implies$ strong existence and uniqueness in law.” Conversely, for certain dual theorems, strong solution plus uniqueness in law may imply pathwise uniqueness (Rehmeier, 2018).

3. Extensions to Multidimensional, Jump, and Infinite-Dimensional Models

a) Multidimensional SDEs. The classical Engelbert–Watanabe–Yamada theorem for scalar SDEs requires the $\sigma$-coefficient to satisfy a “Hölder plus Osgood/Engelbert integrability” condition and the drift to be (locally) Lipschitz. Multidimensional analogues (see Graczyk–Małecki and Lyappieva–Veretennikov) require componentwise diagonal-diffusion structures, possibly only Borel measurable in certain coordinates, combined with Osgood-type modulus and Zvonkin transformations to regularize irregular drifts. Under these, strong uniqueness is restored in higher dimensions, including particle systems, matrix diffusions, and non-colliding ensembles (Graczyk et al., 2011, Lyappieva et al., 5 Oct 2025).

b) Equations with jumps. For SDEs driven by both Brownian motion and a Poisson random measure,

$$dX_t = b(t, X_t)\,dt + \sigma(t, X_t)\,dW_t + \int f(t, X_{t-},u)\,\tilde{N}(dt, du) + \int g(t,X_{t-},u)\,N(dt, du),$$

analogues of the Yamada–Watanabe principle hold: pathwise uniqueness plus weak existence implies strong existence, with careful use of canonical representations on product spaces, measurable selection arguments, and Poisson point process machinery (Barczy et al., 2013).

c) SPDEs: Hilbert and Banach Space Theory. The theorem extends to infinite-dimensional settings such as SPDEs in Gelfand triples or Banach spaces. The equivalence of weak existence plus pathwise uniqueness to strong existence plus uniqueness in law is preserved for mild, variational, and analytically weak solutions, with specific requirements on the geometry (martingale type 2, UMD) and detailed stochastic integration theory (Theewis, 31 Jan 2025, Rehmeier, 2018).

4. Methodological Ingredients and Proof Architecture

The proof constructions depend on:

• Disintegration and canonical representation: Any admissible law can be written as a mixture of conditional laws, allowing measurable selectors to be constructed. In strong cases, the selector must ignore auxiliary randomness and depend only on input $Y$.
• Compatibility and coupling: Canonical couplings constructed with independent extra randomness preserve the compatibility structure.
• Transformation and reduction: For irregular coefficients, Zvonkin’s transform (via ODE/PDE correspondences) can regularize the system to a form where Osgood or Engelbert conditions apply (Lyappieva et al., 5 Oct 2025).
• Bihari–LaSalle and Tanaka–Le Gall methods: For uniqueness, zeroing local times and Osgood-type integral tests are central.

In the SPDE and Banach space context, Kurtz’s measurable stochastic integral representation ensures the well-posedness and universality of the key map $I$, which reconstructs the stochastic integral as a Borel function of data and noise (Theewis, 31 Jan 2025).

5. Dual and Converse Results

In certain infinite-dimensional settings, a “dual” Yamada–Watanabe theorem holds: strong existence plus (joint) uniqueness in law implies pathwise uniqueness. This duality is established for variational solutions of SPDEs:

• Existence of a strong solution and joint uniqueness in law $\implies$ pathwise uniqueness (Rehmeier, 2018).
• Uniqueness in law and joint uniqueness in law are equivalent for deterministic initial data.
• The structure and regularity required for uniqueness may now be distributed among different solution and noise components in the infinite-dimensional case.

6. Applications and Specializations

Applications include:

• Itô diffusions: The original theorems of Yamada–Watanabe (1971), and Engelbert’s extension, for SDEs with irregular (possibly Hölder or weaker) coefficients (Ning et al., 15 Dec 2025).
• Semimartingale-driven SDEs: The compatibility structure allows handling of general driver processes, with martingale-preserving filtration extensions (Kurtz, 2013).
• Particle systems and random matrices: The Graczyk–Małecki multidimensional and spectral theorems resolve strong uniqueness for eigenvalue and eigenvector processes in matrix-valued diffusions, squared Bessel, Wishart, Jacobi processes, and their $\beta$-extensions, critically relying on spectral commutativity and non-collision phenomena (Graczyk et al., 2011).
• Time change, backward SDEs, McKean–Vlasov, and financial models: The compatibility structure encompasses models where the randomness is induced by stopping times, law-dependent coefficients, or Girsanov transforms. In transformed CKLS/CIR/OU models, the theorem rigorously justifies the change of measure and positivity of the process, essential for financial modeling (Ning et al., 15 Dec 2025).

7. Further Generalizations and Notable Corollaries

The abstract Yamada–Watanabe–Engelbert theorem, especially in the Kurtz formulation, unifies earlier scattered results into a single axiomatization applicable in a wide variety of modern settings: Banach space SPDEs (recovering and extending the key results of Ondřeját, Kunze, and Röckner–Schmuland–Zhang), fractional Sobolev spaces, path-dependent coefficients, and non-Markovian stochastic systems (Theewis, 31 Jan 2025). The theorem does not require convexity in law for many McKean–Vlasov-type systems and supports advanced stochastic analysis applications requiring delicate pathwise and distributional uniqueness properties. The introduction of measurable representations of the stochastic integral in UMD or martingale-type 2 Banach spaces creates new tools for encoding solution concepts and facilitates extension to progressively more general stochastic modeling paradigms.