Yamada–Watanabe theorem


The Yamada–Watanabe theorem is a result from probability theory saying that for a large class of stochastic differential equations a weak solution with pathwise uniqueness implies a strong solution and uniqueness in distribution. In its original form, the theorem was stated for -dimensional Itô equations and was proven by Toshio Yamada and Shinzo Watanabe in 1971. Since then, many generalizations appeared particularly one for general semimartingales by Jean Jacod from 1980.

Yamada–Watanabe theorem

History, generalizations and related results

Jean Jacod generalized the result to SDEs of the form
where is a semimartingale and the coefficient can depend on the path of.
Further generalisations were done by Hans-Jürgen Engelbert and Thomas G. Kurtz. For SDEs in Banach spaces there is a result from Martin Ondrejat, one by Michael Röckner, Byron Schmuland and Xicheng Zhang and one by Stefan Tappe.
The converse of the theorem is also true and called the dual Yamada–Watanabe theorem. The first version of this theorem was proven by Engelbert and a more general version by Alexander Cherny.

Setting

Let and be the space of continuous functions. Consider the -dimensional Itô equation
where

Basic terminology

We say uniqueness in distribution, if for two arbitrary solutions and defined on filtered probability spaces and, we have for their distributions, where.
We say pathwise uniqueness if any two solutions and, defined on the same filtered probability spaces with the same -Brownian motion, are indistinguishable processes, i.e. we have -almost surely that

Theorem

Assume the described setting above is valid, then the theorem is:
Jacod's result improved the statement with the additional statement that