a cadlag martingale
/ cah-dlag (French: continu à droite, limite à gauche) /
A martingale in continuous time is an adapted, integrable process (M_t) with E[M_t given F_s] = M_s for all s <= t — the fair-game property of Vol I, now indexed by a continuous parameter. The subtlety is that this defines M only as a collection of equivalence classes of random variables, one per time t, with no control on how the path behaves between or at times. Cadlag is the canonical choice of path regularity: a function is cadlag if it is right-continuous everywhere and has a left limit at every point (French: continu à droite, limite à gauche). A cadlag martingale is a martingale whose almost-every sample path is a cadlag function of t.
The reason cadlag is the right notion, not full continuity, is twofold. First, many fundamental martingales genuinely jump — the compensated Poisson process N_t - lambda t, or the value process of a pure-jump Levy martingale — yet are perfectly good martingales; demanding continuity would exclude them. Second, Doob's regularization theorem guarantees that under the usual conditions, every martingale (and every supermartingale with right-continuous expectation) has a cadlag modification, so cadlag is not a loss of generality but the natural canonical version. Working with the cadlag version makes the supremum sup_{s <= t} |M_s| measurable, makes hitting times stopping times, and gives a well-defined left-limit process M_{t-} and jump process Delta M_t = M_t - M_{t-} that drive the jump terms in Ito's formula.
Cadlag is the standing path assumption for essentially all of stochastic calculus: semimartingales, the optional and predictable quadratic variations, stochastic integrals, and SDE solutions are all built on cadlag (often continuous) integrators. The caveat is that 'a martingale has a cadlag modification' requires the usual conditions on the filtration and, for the right-continuity of paths, the right-continuity of t -> E[M_t] (automatic for martingales since that map is constant). Two cadlag processes that agree almost surely at each fixed time and are both cadlag are indistinguishable — their paths coincide for almost every omega — which is why cadlag upgrades the weak 'modification' notion to the strong 'indistinguishable' notion.
The compensated Poisson process M_t = N_t - lambda t is a cadlag martingale: each path is constant between arrival times, jumps up by 1 at each arrival (right-continuous, with a left limit one unit below), and has E[M_t given F_s] = M_s. Its jump process is Delta M_t = 1 at each arrival and 0 otherwise.
A pure-jump martingale: cadlag, not continuous — exactly why we demand cadlag rather than continuity.
Two distinct cadlag processes can have the same finite-dimensional distributions yet different path properties; conversely, two cadlag modifications of one martingale are indistinguishable, which is the strong sense in which 'the' cadlag version is unique.