Why do we need optional stopping theorem?

Question

For martingale,optional stopping theorem states:

Let $(M_n)_{n\in \mathbb{N}}$ be adapted with $M_n\in L^1$ for all $n$ and if $(M_n)_{n\in \mathbb{N}}$ is a martingale, then $E[M_T]=E[M_0]$, for all bounded stopping time T.

So, why do we need this theorem? If $(M_n)_{n\in \mathbb{N}}$ is a martingale, then by its property $E[M_n]=E[M_i]=E[M_0], \forall i<n$. So $E[M_T]=E[M_0]$ will be implied.

Have I misunderstood something here?

Answer 1

The stopping time $\tau$ can take multiple values of $n$, so we have, when $\tau$ is bounded by $k$ that $$\mathbb{E} M_\tau = \mathbb{E} \sum_{n=1}^k M_n \mathbf{1}_{\{\tau = n\}}= \sum_{n=1}^k \mathbb{E} M_n \mathbf{1}_{\{\tau = n\}}.$$ And we do not know whether $\sum_{n=1}^k \mathbb{E} M_n \mathbf{1}_{\{\tau = n\}} = \mathbb{E} M_0$ without the optional stopping theorem.