= Solution
First stop the solution on leaving $[-n,n]$, at $\tau_n$, and set $F_n(t)=\mathbb E\sup_{u\leq t}|X_{u\wedge\tau_n}|^2$. The <stochastic integral> representation, the <Doob L2 maximal inequality> and the <Cauchy-Schwarz inequality> give, for $0\leq t\leq1$,
$$
\begin{aligned}
F_n(t)&\leq 2\mathbb E\sup_{u\leq t}\left|\int_0^{u\wedge\tau_n}\sigma(X_s)\,dB_s\right|^2
+2\mathbb E\sup_{u\leq t}\left|\int_0^{u\wedge\tau_n}b(X_s)\,ds\right|^2\\
&\leq 8\mathbb E\int_0^{t\wedge\tau_n}\sigma(X_s)^2\,ds
+2t\mathbb E\int_0^{t\wedge\tau_n}b(X_s)^2\,ds\\
&\leq8A\int_0^t(1+F_n(s))\,ds.
\end{aligned}
$$
Here stopping ensures square integrability before the calculation, and $8\sigma^2+2t b^2\leq8(\sigma^2+b^2)$ explains the constant. The <Gronwall inequality> gives $F_n(t)\leq e^{8At}-1$. For globally <Lipschitz functions> as coefficients, the <global existence theorem for stochastic differential equations with Lipschitz coefficients> supplies a nonexplosive solution. Letting $n$ increase and using <Fatou's lemma> proves the slightly stronger <maximal second-moment bound under linear growth>:
$$
\boxed{\mathbb E\sup_{t\leq1}|X_t|^2\leq e^{8A}-1\leq e^{8A}.}
$$
For coefficients that are only <locally Lipschitz functions>, let $\pi_n(x)=\max(-n,\min(x,n))$ and replace the coefficients by $\sigma(\pi_n(x))$ and $b(\pi_n(x))$. These are globally <Lipschitz functions>, agree with the originals on $[-n,n]$, and obey the same <linear growth condition for an SDE>, since $|\pi_n(x)|\leq|x|$. The globally defined solutions agree until their common exits from $[-n,n]$ by <pathwise uniqueness>. The uniform preceding estimate and <Markov's inequality> imply
$$
\mathbb P(\tau_n\leq1)\leq\frac{e^{8A}-1}{n^2}\longrightarrow0.
$$
Patch the solutions before their increasing exit times. Their limiting lifetime exceeds $1$ almost surely by this bound. \b[Thus a pathwise unique strong solution exists throughout $[0,1]$ even for locally Lipschitz coefficients.]
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