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 Shinzō Watanabe in 1971.[1] Since then, many generalizations appeared particularly one for general semimartingales by Jean Jacod from 1980.[2]

Yamada–Watanabe theorem

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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]).

Setting

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

 

where

  •   and   are predictable processes,
  •   is an  -dimensional Brownian Motion,
  •   is deterministic.

Basic terminology

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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  

Theorem

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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]

References

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  1. ^ Yamada, Toshio; Watanabe, Shinzō (1971). "On the uniqueness of solutions of stochastic differential equations". J. Math. Kyoto Univ. 11 (1): 155–167. doi:10.1215/kjm/1250523691.
  2. ^ a b c Jacod, Jean (1980). "Weak and Strong Solutions of Stochastic Differential Equations". Stochastics. 3: 171–191. doi:10.1080/17442508008833143.
  3. ^ a b c Engelbert, Hans-Jürgen (1991). "On the theorem of T. Yamada and S. Watanabe". Stochastics and Stochastic Reports. 36 (3–4): 205–216. doi:10.1080/17442509108833718.
  4. ^ Kurtz, Thomas G. (2007). "The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities". Electron. J. Probab. 12: 951–965. doi:10.1214/EJP.v12-431.
  5. ^ Ondreját, Martin (2004). "Uniqueness for stochastic evolution equations in Banach spaces". Dissertationes Math. (Rozprawy Mat.). 426: 1–63.
  6. ^ Röckner, Michael; Schmuland, Byron; Zhang, Xicheng (2008). "Yamada–Watanabe theorem for stochastic evolution equations in infinite dimensions". Condensed Matter Physics. 11 (2): 247–259.
  7. ^ Tappe, Stefan (2013), "The Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations", Electronic Communications in Probability, 18 (24): 1–13
  8. ^ a b Cherny, Alexander S. (2002). "On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations". Theory of Probability & Its Applications. 46 (3): 406–419. doi:10.1137/S0040585X97979093.