= Solution
Write the diffusion and <drift coefficients> as
$$
\sigma(x)=\frac1{1+x^2},\qquad b(x)=-\frac1{2(1+x^2)^2}.
$$
Both are continuously differentiable, with
$$
\sigma'(x)=-\frac{2x}{(1+x^2)^2},\qquad
b'(x)=\frac{2x}{(1+x^2)^3}.
$$
These derivatives are bounded. For example $2|x|\leq1+x^2$ bounds both absolute values by one. The <mean value theorem> therefore makes both coefficients <globally Lipschitz>. They are also bounded, so satisfy the <linear growth condition for an SDE>.
The <global existence theorem for stochastic differential equations with Lipschitz coefficients> consequently gives
$$
\boxed{\text{a global strong solution, unique up to indistinguishability}.}
$$
In particular, there is no finite explosion time. The solution has continuous paths, so all subsequent level-hitting and stopping arguments apply.
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