a local martingale
A local martingale is a process that is a martingale 'locally in time' — it becomes a true martingale once you stop it at a suitable sequence of stopping times that grows to infinity, even if the un-stopped process fails the integrability needed to be a genuine martingale. Local martingales are the natural and indispensable enlargement of the class of martingales for stochastic calculus: stochastic integrals against martingales, and the martingale part of a general semimartingale, are local martingales but typically not true martingales, so the whole theory must be phrased at this level.
Definition: a cadlag adapted process M with M_0 = 0 is a local martingale if there exists a sequence of stopping times T_n increasing to infinity (a localizing or reducing sequence) such that for each n the stopped process M^{T_n}_t = M_{t and T_n} is a uniformly integrable (true) martingale. The point of localization is to defeat integrability failures: even if E[|M_t|] = infinity for the raw process, each stopped piece is integrable and fair. Every (true) martingale is a local martingale (take T_n = n), but the converse is false — and this gap is one of the most important honest facts in the subject. A local martingale that is bounded below (or is of class DL, or whose family {M_T : T <= n} is UI for each n) is a true martingale; in particular a nonnegative local martingale is a supermartingale, hence E[M_t] <= E[M_0], with equality (true-martingale behavior) iff the supermartingale is constant in mean.
Why it matters and the central caveat: a strict local martingale is a local martingale that is NOT a true martingale, and these genuinely exist. The classic example is the inverse-Bessel-3 process, or equivalently 1/|B| for a 3-dimensional Brownian motion away from 0 (or its reciprocal-type relatives): it is a positive local martingale with E[M_t] strictly decreasing in t, so E[M_t] < E[M_0] — the martingale property fails despite local fairness. This is not a pathology to be avoided; it is why option-pricing 'bubbles', the difference between weak and strong solutions of SDEs, and the failure of E[stochastic exponential] = 1 (Novikov's condition is exactly the patch) all hinge on distinguishing local from true martingales. Never assume a local martingale has constant expectation.
Let B be a 3-dimensional Brownian motion started away from the origin and set M_t = 1/|B_t|. Ito's formula shows M is a local martingale (1/|x| is harmonic in R^3 off the origin), and it is positive, hence a supermartingale; but E[M_t] strictly decreases, so M is a strict local martingale — local but not true.
The inverse-Bessel-3 process: a positive local martingale whose mean strictly decreases — the canonical strict local martingale.
A local martingale need not be a true martingale; the safe sufficient conditions are boundedness below plus the supermartingale-with-constant-mean criterion, domination by an integrable variable, or class DL — never assume constant expectation for free.