Class D process ($\{X_{\tau }:\tau \le T\text{ a stopping time}\}\text{ is uniformly integrable}$)

On a finite time horizon, an adapted process is of class D if its values at all stopping times in that horizon form a uniformly integrable family. Deterministic boundedness implies this property. The stopping-time uniformity is stronger than having an integrable value at each fixed time.

An adapted process is of class DL when, for every finite $T$, the family $\{X_{\tau }:\tau \le T\text{ a stopping time}\}$ is uniformly integrable. A local martingale is a true martingale exactly when it has this property. This is a local-in-time condition, not uniform integrability over the whole half-line.

// Target: probability-and-statistics.bigb