= Solution
Let $X,Y$ be two solutions with the same Brownian driver and the same initial value. Write $D=X-Y$, and take a common <Lipschitz> constant $L$ for both coefficients. Stop when $|X|+|Y|$ reaches $n$. Before this <stopping time> $\tau_n$, all difference integrands below are bounded. The <Itô formula> gives
$$
d(D^2)=2D[\sigma(X)-\sigma(Y)]\,dB
+\left([\sigma(X)-\sigma(Y)]^2+2D[b(X)-b(Y)]\right)dt.
$$
The stopped <stochastic integral> is a true <martingale> on every finite horizon. <Lipschitz continuity> therefore yields
$$
\mathbb E D_{t\wedge\tau_n}^2
\le(L^2+2L)\int_0^t\mathbb E[\mathbf1_{\{s<\tau_n\}}D_s^2]ds
\le(L^2+2L)\int_0^t\mathbb E D_{s\wedge\tau_n}^2ds.
$$
Since $D_0=0$, the <Gronwall inequality> makes the left side zero. Let $n\to\infty$; the continuous global solutions are bounded on each compact time interval, so their <stopping times> exhaust it. Equality first at every rational time and then by continuity gives indistinguishability. This proves \b[<pathwise uniqueness> for globally <Lipschitz> drift and diffusion coefficients]. The argument does not require the initial value to have a global second moment: after stopping, the difference is bounded and initially zero.
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