= Maximal second-moment bound under linear growth
{title2=$\mathbb E\sup_{s\leq t}|X_s|^2\leq e^{8At}-1$}
For $dX=\sigma(X)dB+b(X)dt$, $X_0=0$, and $\sigma^2+b^2\leq A(1+X^2)$, the <Doob L2 maximal inequality> and the <Cauchy-Schwarz inequality> imply, for $t\leq1$, $F(t)\leq8A\int_0^t(1+F(s))ds$, where $F(t)=\mathbb E\sup_{s\leq t}|X_s|^2$. The <Gronwall inequality> yields $F(t)\leq e^{8At}-1$. Apply the calculation first to stopped paths; truncating <locally Lipschitz functions> and using this uniform bound gives exit probabilities at most $(e^{8A}-1)/n^2$, proving nonexplosion on the unit time interval.
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