Doob's continuous-time regularity theorems
/ Doob = doob /
When you pass from discrete to continuous time, the first job is to make sense of paths. Given only the defining identity E[M_t given F_s] = M_s, a martingale is a family of L^1 random variables; nothing yet says the path t -> M_t(omega) is even measurable, let alone continuous or limit-having. Doob's regularity (or regularization) theorems are the package of results that, under the usual conditions, manufacture a good path version and then describe how it behaves and converges. They are the continuous-time analogues of the discrete Doob convergence theorems, and they are what let continuous-time martingale theory get off the ground.
The core results are four. Regularization: every martingale (more generally every supermartingale with right-continuous t -> E[X_t]) admits a cadlag modification once the filtration satisfies the usual conditions; the proof rests on Doob's upcrossing inequality applied along the rationals to bound the number of oscillations across any band, forcing right and left limits to exist along rationals, which are then declared the cadlag path. Almost-sure convergence: an L^1-bounded cadlag (sub/super)martingale converges almost surely to an integrable limit (the upcrossing argument again). L^1 / closure convergence: a cadlag martingale converges in L^1 and is closed if and only if it is uniformly integrable. L^p convergence and Doob's maximal inequality: for p > 1, an L^p-bounded martingale converges in L^p and satisfies E[(sup_{s <= t} |M_s|)^p] <= (p/(p-1))^p E[|M_t|^p], controlling the whole running maximum by the terminal moment.
These theorems are the reason one may always assume a martingale is cadlag, may take limits of martingales, and may bound suprema of paths — prerequisites for optional stopping, stochastic integration, and the Doob-Meyer decomposition. The honest caveats: the maximal inequality with the (p/(p-1)) constant needs p > 1 and genuinely fails at p = 1 (there one has only the weak-type inequality lambda P(sup |M_s| > lambda) <= E[|M_t|]); L^1-boundedness gives only a.s. convergence, not L^1 convergence; and right-continuity of paths requires the usual conditions on the filtration, without which no cadlag version need exist.
Doob's L^2 maximal inequality applied to Brownian motion gives E[(sup_{s <= t} B_s)^2] <= 4 E[B_t^2] = 4t, instantly controlling the running maximum of the path by the terminal variance — a typical use that makes pathwise estimates from one-time-point moments.
The maximal inequality (constant (p/(p-1))^p, here p=2 gives 4) turns a terminal moment bound into a bound on the whole path.
The clean L^p maximal inequality needs p > 1; at p = 1 it degrades to a weak-type inequality, and there is the related L log L bound — a frequent source of error is quoting the (p/(p-1)) form at p = 1.