= Solution
The <existence and pathwise uniqueness theorem for a stochastic differential equation> states that <locally Lipschitz> drift $b$ and diffusion $\sigma$ on an open domain give a pathwise unique <strong solution of a stochastic differential equation> up to a maximal lifetime $\xi$. The initial value is measurable with respect to the initial sigma-algebra, and the filtration satisfies the <usual conditions for a filtration>. On $\mathbb R^d$, if $T_n$ are exits from increasing balls, then $T_n\uparrow\xi$; a finite lifetime entails escape from every compact subset. The coefficients need not satisfy a global growth bound for this local result.
One construction truncates the coefficients outside each compact region to <globally Lipschitz> coefficients, solves the resulting global equations, and patches their solutions until the exits. <Pathwise uniqueness> makes solutions from larger truncations agree on the smaller stopped interval. Their union is the <maximal local solution of a stochastic differential equation>, and any extension would contradict the exhaustion of the domain by the exit times.
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