the localization technique
Localization is the standard proof method of stochastic calculus for extending results from a well-behaved class (bounded, or square-integrable, or true martingales) to a much larger class (local martingales, semimartingales, locally integrable processes) by chopping time with stopping times. The idea: prove what you want for nice processes, then approximate a general process by stopping it at times T_n that grow to infinity, so that each stopped piece is nice, and finally let n -> infinity. It is the lever that turns the bounded theory into the full theory.
The mechanism: a property is said to hold locally if there is a sequence of stopping times T_n increasing to infinity such that the stopped process X^{T_n} has the property for each n. Typical localizing sequences are the exit times of growing balls, T_n = inf{t : |X_t| >= n}, or first times the quadratic variation exceeds n, T_n = inf{t : [X]_t >= n}; because X is cadlag and these thresholds grow, T_n increases to infinity almost surely. One proves, say, the Ito isometry or the existence of <M> for bounded or square-integrable martingales, then applies it to each M^{T_n}, then checks the pieces are consistent (the objects agree on overlaps) and passes to the limit. The same scheme defines locally square-integrable martingales, locally bounded predictable integrands, and the stochastic integral against a general local martingale or semimartingale.
Localization is why the definitions of local martingale, the stochastic integral, the quadratic variation of an arbitrary continuous local martingale, and the Doob-Meyer compensator of a locally-integrable submartingale all make sense. The honest caveats are two. First, localization is not free at the limit: many properties survive localization (being a local martingale, having a quadratic variation) but the true-martingale property does NOT — stopping a strict local martingale gives true martingales at each stage yet the limit can fail to be a true martingale (this is exactly the strict-local-martingale phenomenon). Second, you must verify the T_n really do increase to infinity (a non-explosion check); if the localizing times accumulate at a finite explosion time, the localization only describes the process up to explosion.
To define the quadratic variation of an unbounded continuous local martingale M, set T_n = inf{t : |M_t| >= n}. Each M^{T_n} is a bounded martingale, so it has a well-defined <M^{T_n}>; these agree on [0, T_n], and since T_n increases to infinity one glues them into a single increasing process <M> defined for all t.
Localizing by exit times reduces an unbounded problem to a sequence of bounded ones, then glues the answers.
Localization preserves being a local martingale and the bracket processes, but it does not recover the true-martingale property at the limit — that is precisely how strict local martingales arise from a sequence of genuine martingales.