= Solution
Relative to a <filtration> satisfying the usual conditions, a <local martingale> is an <adapted> <càdlàg process> $X$ for which there exist increasing <stopping times> $S_n\uparrow\infty$ almost surely such that every stopped process $X^{S_n}_t=X_{t\wedge S_n}$ is an integrable <martingale>. Thus, for $s\le t$,
$$
\mathbb E[X_{t\wedge S_n}\mid\mathcal F_s]=X_{s\wedge S_n}.
$$
The sequence is a <localizing sequence>. A <continuous local martingale> additionally has continuous sample paths. The usual definition entails integrability of the initial value; <local martingales> need not be integrable at later times without stopping.
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