Abstract
Assume that the weak existence and pathwise uniqueness hold for solutions to the equation $dX_t=\sigma(t,X)dB_t + b(t,X)dt$ starting at fixed points. then there exists a Borel measurable function F, such that any solution (X,B) satisfies $X = F(X_0,B)$ a.s. This strengthens a fundamental result of Yamada and Watanbe, where F may depend on the initial distribution $\mu$
Citation
Olav Kallenberg. "On the existence of universal functional solutions to classical SDE's." Ann. Probab. 24 (1) 196 - 205, January 1996. https://doi.org/10.1214/aop/1042644713
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