Solving SDEs in a pathwise manner?
Solution 1:
The existence of the functional $\Phi$ is often taken as a definition of a strong solution. Your question boils down to "when does a strong solution exist?". One thing to look up is the classic result of Yamada-Watanabe by which weak existence and pathwise uniqueness imply strong existence and the uniqueness in probablility. See for example the book Karatzas & Shreve, Brownian Motion and Stochastic Calculus or the papers [1] and [2].