 Solution

Use the Itô formula up to $T_n$:For fixed $n$, the stopped stochastic integrand is bounded on each finite horizon, so its integral has expectation zero. The linear growth condition for an SDE and $2xb(x)\le x^2+b(x{)}^2$ giveWriting $u_n(t)=\mathbb{E}{X}_{t\wedge T_n}^2$, and bounding the active-time integrand by the stopped square, yieldsThe Gronwall inequality gives a bound independent of $n$:On $\{T_n\le t\}$, continuity gives $|X_{t\wedge T_n}|=n$. The Markov inequality therefore givesSince $T_n\uparrow \xi$, the event $\{\xi \le t\}$ is the intersection of these decreasing events and has probability zero. Taking integer $t$ proves $\xi =\infty$ almost surely. This proves global existence with locally, rather than globally, Lipschitz coefficients under the stated growth bound.