Existence and non-existence of solutions of one-dimensional stochastic equations

We consider the one-dimensional stochastic equation [formula] for a continuous local martingale M with square variation [M] and measurable drift and diffusion coefficients b and σ. The main purpose of this paper is to derive a necessary condition for the existence of a solution X starting from x0. As a result, we construct a diffusion coefficient σ such that the above stochastic equation has no solution X whatever the initial value x0 and the non-zero, say, continuous drift coefficient b might be.