= Local solution of a stochastic differential equation
A local solution is a <locally defined stochastic process> taking values in a specified open domain $U$ and satisfying the <stochastic differential equation> on every stopped interval before its lifetime. For $dX=b(X)dt+\sigma(X)dB$, the integral equation holds after each announcing stop, with the requisite local drift and noise integrability. A <maximal local solution of a stochastic differential equation> cannot be extended while remaining in $U$; a finite lifetime can be a boundary hit rather than divergence to infinity.
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