On the existence of universal functional solutions to classical SDE's (Q1922073)

The paper gives an interesting improvement of the following fundamental result of \textit{T. Yamada} and \textit{S. Watanabe} [J. Math. Kyoto Univ. 11, 155-167 (1971; Zbl 0236.60037) and ibid. 11, 553-563 (1971; Zbl 0229.60039)]. For a classical stochastic differential equation in \(R^d\) of the form \(dX_t= \sigma(t,X)dB_t+b(t,X)dt\), weak existence and pathwise uniqueness for a given initial distribution \(\mu\) imply strong existence and uniqueness in law for that \(\mu\), and the existence of a Borel measurable function \(F_\mu:R^d\times C(R_+,R^r)\to C(R_+,R^d)\) such that \(X=F_\mu(X_0,B)\) holds a.s. for every weak solution \((X,B)\) with initial distribution \(\mu\). The author proves that a universal representation \(X=F(X_0,B)\) (i.e. with \(F\) independent of \(\mu\)) can be obtained. Precisely, under the assumption that weak existence and pathwise uniqueness hold for all initial distributions \(\delta_X\), there exists a Borel measurable and universally predictable function \(F\) such that \(X=F(X_0,B)\) a.s. for every weak solution \((X,B)\).