Local martingale

Remarks

Every càdlàg martingale $X$ is a local martingale: take $\tau_n = n$, since a stopped martingale is a martingale by optional stopping. The deterministic process $X_t = t$ is adapted and continuous but is not a local martingale. Indeed, clause 3 would give $E[t \wedge \tau_n] = 0$ for all $t$, so $\tau_n = 0$ almost surely, contradicting clause 2. A local martingale need not be a martingale: if $B$ is a three-dimensional Brownian motion started at $x \ne 0$, then $1/\abs{B_t}$ is a local martingale but not a martingale. A non-negative local martingale is a supermartingale, by Fatou's lemma. Some authors require clause 3 only for $X^{\tau_n} - X_0$, or for $X^{\tau_n} \mathbf{1}\{\tau_n > 0\}$, so that $X_0$ need not be integrable.