Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion.
I am trying to show that $X_t$ cannot blow up in finite time (this is an exercise in Karatzas and Shreve).
Does anyone have any ideas on this one. I can't get anywhere. Thanks
We have $X_{t \wedge S_n} = \int_0^{t \wedge S_n} \sigma(X_s)dW_s$. Now letting $n \rightarrow \infty$, we have on the right side that the limit is either finite or does not exist, depending on whether or not the quadratic variation is finite or infinite (this can be seen from a time change argument in which we time change to BM and then apply Law of Iterated Logarithm).
On the left the limit must be infinite on the set $\{S<t\}$ and so the measure of this set is zero since there are no infinite limits on the right. Since this is true for all $t$, we have $P[S=\infty] = 1$.