I'm working with the book "Continuous Martingales and Brownian Motion", it is used frequently the next result (which is in the appendix):
But I'm not sure how they get to the equation $(+)$, using integration by parts formula, I obtained the formula with $g(y)=\frac{\phi'_\mu (y)}{\phi_\mu(y)}$ but instead of $1$ with $\frac{\phi'_\mu (b)}{\phi_\mu(b)}$. Anyway they mentioned that the existence of a solution (in a bounded interval) for $(+)$ follows from the locally Lipschitz condition of the map $x\to x^2$, but I'm not sure that every positive radon measure $\mu$ is locally Lipschitz, because in general I think is not true that the second (left -for example-) derivate of a convex function is locally bounded.
Do you know where is my mistake? Do you know any other reference for this equation?