= Solution
A <locally defined stochastic process> is a pair $(X,T)$, where $T$ is a lifetime <stopping time> and $X_t(\omega)$ is defined for $0\leq t<T(\omega)$, that is, on a <stochastic interval>. For continuous local problems one uses <stopping times> $T_n\uparrow T$ with $T_n<T$ on $\{T>0\}$, so the process stopped at each $T_n$ has an ordinary continuous <adapted process> version. This is an <announcing sequence for a stopping time>; no value at $T$ or after $T$ is implicit in the pair.
A <local solution of a stochastic differential equation> on an open domain $U$ takes its values in $U$ before $T$ and, on every such stopped interval, satisfies
$$
X_{t\wedge T_n}=X_0+\int_0^{t\wedge T_n}b(X_s)\,ds+\int_0^{t\wedge T_n}\sigma(X_s)\,dB_s,
$$
with the local integrability needed for the ordinary and <Itô integrals>. A <maximal local solution of a stochastic differential equation> has no extension to a strictly later lifetime that agrees with it before $T$. For locally <Lipschitz continuous> coefficients on $U$, take a nested <compact exhaustion> $K_n$ of $U$ and the successive exit times: their increasing limit is the maximal lifetime, and on $\{T<\infty\}$ the solution eventually leaves every compact subset of $U$. To obtain the announcing sequence with finite stops even when the path never exits a compact set, cap these exit times by $n$; the capped times still increase to the maximal lifetime. Thus \b[a finite maximal lifetime means exit from the domain or explosion; it need not mean divergence to infinity]. For $U=(0,\infty)$, reaching the boundary zero terminates the local solution even if an absorbing extension could be defined for a different domain.
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