Let $W_t$ be a Brownian Motion, $X_t$ and $M(t)$ be two stochastic processes: $$X_t=adt+bdW_t$$ $$M(t)=e^{-\lambda t}\Phi(X_t)$$ where $\Phi$ is a deterministic function determined from the requirement that $M_t$ is a martingale and $X_t$ assumed to have values in the interval $[L,U]$.
By applying Ito's lemma and fixing the zero drift, we get $$\lambda\Phi(X_t)=a\Phi^\prime (X_t)+\frac{b^2}{2}\Phi^{\prime\prime}(X_t)$$ which can be represented by the ODE $\lambda\Phi(x)=a\Phi^\prime(x)+\frac{b^2}{2}\Phi^{\prime\prime}(x)$, which has two linearly independent solutions, $\phi_1(x)$ and $\phi_2(x)$, yielding two local martingales: $M_{1,2}(t)=e^{-\lambda t}\phi_{1,2}(X_t)$.
What else do I need in order to get that $M_{1,2}(t)$ are global martingales? Suppose that $X_t$ does not have any singular point. Does this suffice to get that $M_{1,2}(t)$ are bounded?