= Solution
Take a complete, right-continuous <filtration> $(\mathcal F_t)$, an increasing family of sigma-algebras. Adapted means that $X_t$ is $\mathcal F_t$-measurable at each $t$; <càdlàg> means right-continuous with left limits. A <local martingale> is an adapted <càdlàg> process $X$, with integrable initial value, for which there are <stopping times> $\tau_n\uparrow\infty$ almost surely such that each <stopped process> $X^{\tau_n}_t=X_{t\wedge\tau_n}$ is a true <martingale>. Such a sequence is a localizing sequence. A true <martingale> is integrable at each time and satisfies $\mathbb E(X_t\mid\mathcal F_s)=X_s$ for $s\le t$.
The necessary and sufficient condition is that $X$ is a <Class DL process>:
$$
\boxed{X\text{ is a martingale}\ \Longleftrightarrow\ \{X_\tau:\tau\le T\text{ a stopping time}\}\text{ is uniformly integrable for every finite }T.}
$$
A family $\mathcal Z$ is <uniformly integrable> if its members are integrable and $\sup_{Z\in\mathcal Z}\mathbb E(|Z|\mathbf1_{\{|Z|>K\}})\to0$ as $K\to\infty$. A <stopping time> $\tau$ satisfies $\{\tau\le t\}\in\mathcal F_t$ for all $t$.
For necessity, on $[0,T]$ the <optional stopping theorem> gives $X_\tau=\mathbb E(X_T\mid\mathcal F_\tau)$; <conditional expectations> of one integrable variable form a <uniformly integrable> family. For sufficiency, $X_{t\wedge\tau_n}\to X_t$ almost surely, and <Class DL> upgrades this convergence to $L^1$ on each finite horizon. Pass to the limit in $\mathbb E(X_{t\wedge\tau_n}\mid\mathcal F_s)=X_{s\wedge\tau_n}$ to obtain the <martingale> identity. Requiring <uniform integrability> of all times on the entire half-line would be stronger than necessary; <Brownian motion> is already a counterexample to that stronger requirement.
Back to article page
