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On the theorem of T. Yamada and S. Watanabe. (English) Zbl 0739.60046

Summary: The well-known theorem of T. Yamada and S. Watanabe [J. Math. Kyoto Univ. 11, 155-167 (1971; Zbl 0236.60037)] asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0236.60037
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