proof of Martingale criterion (continuous time)

Proof.

1. Let $X$ be a martingale. By the optional sampling theorem we have $E(X_c|{ℱ}_{\tau })=X_{c\wedge \tau }=X_{\tau }\forall \tau \le c$. Since conditional expectations are uniformly integrable the first direction follows.

2. Let ${({\tau }_k)}_{k\ge 1}$ be a local sequence of stopping times (i.e. ${\tau }_k\uparrow \mathrm{\infty }$ a.s. and $X^{{\tau }_k}$ martingale $\forall k\in \mathbb{N}$).For each $t\in {ℝ}_{+}$ we have $X_{{\tau }_k\wedge t}\to X_t,k\to \mathrm{\infty }$ almost surely.The set

$\mathrm{\{}{X}_{{\tau }_k\wedge t}:k\in \mathbb{N}\}$$\subset \{X_{\tau }:\tau \text{stopping time},\tau \le c\}$

is uniformly integrable (take $c=t$). It follows that $X_t^{{\tau }_k}\stackrel{\begin{array}{c}{ℒ}^1\end{array}}{⟶}{X}_t,k\to \mathrm{\infty }$. Since the martingale property is stable under ${ℒ}^1$ convergence, $X$ is a martingale.∎