the Yamada-Watanabe theorem
/ yah-MAH-dah wah-tah-NAH-beh /
The Yamada-Watanabe theorem is the bridge between the weak and strong notions of solution for an SDE. It answers the structural question: how do weak existence, strong existence, uniqueness in law, and pathwise uniqueness relate? The headline consequence is a clean implication: if an SDE has a weak solution AND pathwise uniqueness holds, then it has a (unique) strong solution, and uniqueness in law holds as well. So you can establish strong solvability without ever constructing the solution as a functional of the noise directly — you only need a weak solution (often easy, via the martingale problem or Girsanov) plus a comparison estimate giving pathwise uniqueness.
The precise statement combines two facts. First, pathwise uniqueness implies uniqueness in law (so the law is forced once paths are forced). Second, and this is the deep part, weak existence together with pathwise uniqueness implies the existence of a strong solution: there is a measurable map F such that any solution can be written X = F(X_0, B), i.e. the solution really is a deterministic functional of the initial condition and the driving Brownian path. The proof uses a Skorokhod / Gyongy-Krylov style argument: one builds two weak solutions on a common space driven by the same noise, invokes pathwise uniqueness to force them equal, and extracts from this the measurable solution map. A common modern phrasing is: weak existence + pathwise uniqueness <=> strong existence + uniqueness in law.
This theorem is why practitioners can mix techniques: prove existence weakly (where the analysis is soft) and uniqueness pathwise (where a Lipschitz or Yamada-Watanabe-condition estimate is sharp), and conclude a strong, unique solution. The companion Yamada-Watanabe uniqueness condition handles non-Lipschitz diffusion coefficients: pathwise uniqueness in dimension one holds if |sigma(x) - sigma(y)| <= h(|x - y|) with integral_{0+} h(u)^{-2} du = infinity (e.g. h(u) = sqrt(u), which covers the CIR/Bessel square-root diffusion). The honest caveat: the converse direction is subtle, and pathwise uniqueness is strictly stronger than uniqueness in law — you cannot upgrade uniqueness in law to a strong solution without it (Tanaka's equation is the standing counterexample).
For the CIR process dX = (a - b X) dt + sigma sqrt(X) dB, the square-root diffusion is not Lipschitz at 0, so the standard theorem does not apply. But |sqrt(x) - sqrt(y)| <= sqrt(|x - y|), and h(u) = sqrt(u) satisfies integral_{0+} du/u = infinity, so the Yamada-Watanabe condition gives pathwise uniqueness; with weak existence this yields a unique strong solution.
The Yamada-Watanabe square-root condition rescues the non-Lipschitz CIR diffusion.
The implication runs weak existence + pathwise uniqueness => strong existence; it does NOT run from uniqueness in law. Tanaka's equation has weak existence and uniqueness in law but, lacking pathwise uniqueness, has no strong solution.