This is perhaps a very basic problem regarding stochastic calculus but I am going off the deep end in trying to do some of my homework problems and I think I don't understand some small basic concept. If I have a weak solution to a stochastic differential equations of the form: $$dX_t=a(X_t)dt+\sigma(X_t)dB_t$$ Where $B$ is the Brownian motion, up to a AS finite stopping time $\tau$. Does this imply that $X_\tau$ exists in the extended reals sense? Or it need not be defined? And is whether or not $X_\tau$ defined built into the definition of a weak solution?
The context for this is that I have seen a couple sources say "$X_t$ is continuous in the extended reals sense" (Eg. Karatzas and Shreve) and so $X_\tau$ is either $\pm \infty$. Yet in class we have discussed the fact that (local) martingales of the form $M_t=Y_tdB_t$ can have finite time blowup and thus is not defined at $\tau_\infty$ (blowup time of the quadratic variation of $Y$).