Maximal second-moment bound under linear growth ($\mathbb{E}{\sup }_{s\le t}|X_s{|}^2\le e^{8At}-1$)

For $dX=\sigma (X)dB+b(X)dt$, $X_0=0$, and ${\sigma }^2+b^2\le A(1+X^2)$, the Doob L2 maximal inequality and the Cauchy-Schwarz inequality imply, for $t\le 1$, $F(t)\le 8A{\int }_0^t(1+F(s))ds$, where $F(t)=\mathbb{E}{\sup }_{s\le t}|X_s{|}^2$. The Gronwall inequality yields $F(t)\le e^{8At}-1$. 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.