= Characterization of a martingale by stopped expectations
Let an integrable process $(M_n)$ be <adapted process>[adapted] to $(\mathcal F_n)$. It is a <martingale> if and only if $\mathbb E[M_{n\wedge\tau}]=\mathbb E[M_0]$ for every $n$ and every <stopping time> $\tau$. For the reverse implication, fix $A\in\mathcal F_n$ and take $\tau=n$ on $A$ and $\tau=n+1$ on $A^c$. Comparing its stopped expectation at $n+1$ with the expectation obtained from the deterministic stopping time $n+1$ gives $\mathbb E[\mathbf1_A(M_{n+1}-M_n)]=0$. Since this holds for every $A\in\mathcal F_n$, it is exactly $\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n$.
Back to article page
