A SDE with local lipschitz coefficient

Question

Since $\exp(\cdot)$ is local lipschitz, the following sde has s strong solution $$ \mathrm{d}X_s=\exp(X_s)\mathrm{d}B_s, X(0)=1, $$ where $B$ is a Brownian Motion. I wander if the following expression holds? $$\mathbb{E}\int_{0}^T\exp(2X(s))<\infty.$$