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 ${\displaystyle n}$-dimensional Itô equations and was proven by 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

${\displaystyle dX_{t}=u(X,Z)dZ_{t},}$

where ${\displaystyle (Z_{t})_{t\geq 0}}$ is a semimartingale and the coefficient ${\displaystyle u}$ can depend on the path of ${\displaystyle Z}$.^[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 ${\displaystyle n,r\in \mathbb {N} }$ and ${\displaystyle C(\mathbb {R} _{+},\mathbb {R} ^{n})}$ be the space of continuous functions. Consider the ${\displaystyle n}$-dimensional Itô equation

${\displaystyle dX_{t}=b(t,X)dt+\sigma (t,X)dW_{t},\quad X_{0}=x_{0}}$

where

• ${\displaystyle b\colon \mathbb {R} _{+}\times C(\mathbb {R} _{+},\mathbb {R} ^{n})\to \mathbb {R} ^{n}}$ and ${\displaystyle \sigma \colon \mathbb {R} _{+}\times C(\mathbb {R} _{+},\mathbb {R} ^{n})\to \mathbb {R} ^{n\times r}}$ are predictable processes,
• ${\displaystyle (W_{t})_{t\geq 0}=\left((W_{t}^{(1)},\dots ,W_{t}^{(r)})\right)_{t\geq 0}}$ is an ${\displaystyle r}$-dimensional Brownian Motion,
• ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$ is deterministic.

We say uniqueness in distribution (or weak uniqueness), if for two arbitrary solutions ${\displaystyle (X^{(1)},W^{(1)})}$ and ${\displaystyle (X^{(2)},W^{(2)})}$ defined on (possibly different) filtered probability spaces ${\displaystyle (\Omega _{1},{\mathcal {F}}_{1},\mathbf {F} _{1},P_{1})}$ and ${\displaystyle (\Omega _{2},{\mathcal {F}}_{2},\mathbf {F} _{2},P_{2})}$, we have for their distributions ${\displaystyle P_{X^{(1)}}=P_{X^{(2)}}}$, where ${\displaystyle P_{X^{(1)}}:=\operatorname {Law} (X_{t}^{1},t\geq 0)}$.

We say pathwise uniqueness (or strong uniqueness) if any two solutions ${\displaystyle (X^{(1)},W)}$ and ${\displaystyle (X^{(2)},W)}$, defined on the same filtered probability spaces ${\displaystyle (\Omega ,{\mathcal {F}},\mathbf {F} ,P)}$ with the same ${\displaystyle \mathbf {F} }$-Brownian motion, are indistinguishable processes, i.e. we have ${\displaystyle P}$-almost surely that ${\displaystyle \{X_{t}^{(1)}=X_{t}^{(2)},t\geq 0\}}$

Assume the described setting above is valid, then the theorem is:

If there is pathwise uniqueness, then there is also uniqueness in distribution. And if for every initial distribution, there exists a weak solution, then for every initial distribution, also a pathwise unique strong solution exists.^[3][8]

Jacod's result improved the statement with the additional statement that

If a weak solutions exists and pathwise uniqueness holds, then this solution is also a strong solution.^[2]