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 the Japanese mathematicians Toshio Yamada and Shinzō Watanabe in 1971. [1] Since then, many generalizations appeared particularly one for general semimartingales by Jean Jacod from 1980. [2]
Jean Jacod generalized the result to SDEs of the form
where is a semimartingale and the coefficient can depend on the path of . [2]
Further generalisations were done by Hans-Jürgen Engelbert (1991 [3]) and Thomas G. Kurtz (2007 [4]). For SDEs in Banach spaces there is a result from Martin Ondrejat (2004 [5]), one by Michael Röckner, Byron Schmuland and Xicheng Zhang (2008 [6]) and one by Stefan Tappe (2013 [7]).
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 (1991 [3]) and a more general version by Alexander Cherny (2002 [8]).
Let and be the space of continuous functions. Consider the -dimensional Itô equation
where
We say uniqueness in distribution (or weak uniqueness), if for two arbitrary solutions and defined on (possibly different) filtered probability spaces and , we have for their distributions , where .
We say pathwise uniqueness (or strong 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
Assume the described setting above is valid, then the theorem is:
Jacod's result improved the statement with the additional statement that
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 the Japanese mathematicians Toshio Yamada and Shinzō Watanabe in 1971. [1] Since then, many generalizations appeared particularly one for general semimartingales by Jean Jacod from 1980. [2]
Jean Jacod generalized the result to SDEs of the form
where is a semimartingale and the coefficient can depend on the path of . [2]
Further generalisations were done by Hans-Jürgen Engelbert (1991 [3]) and Thomas G. Kurtz (2007 [4]). For SDEs in Banach spaces there is a result from Martin Ondrejat (2004 [5]), one by Michael Röckner, Byron Schmuland and Xicheng Zhang (2008 [6]) and one by Stefan Tappe (2013 [7]).
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 (1991 [3]) and a more general version by Alexander Cherny (2002 [8]).
Let and be the space of continuous functions. Consider the -dimensional Itô equation
where
We say uniqueness in distribution (or weak uniqueness), if for two arbitrary solutions and defined on (possibly different) filtered probability spaces and , we have for their distributions , where .
We say pathwise uniqueness (or strong 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
Assume the described setting above is valid, then the theorem is:
Jacod's result improved the statement with the additional statement that