Suppose i have $X_1,X_2,…\sim i.i.d. N(0,1)$
$$S_n=X_1+…+X_n$$
filtration $\mathcal{F}_n=\sigma(X_1,…,X_n)$.
$S_n$ is adapted and integrable as finite sum of adapted and integrable random variable.
$$E[S_{n+1}|\mathcal{F}_n]=E[S_{n}|\mathcal{F}_n]+E[X_{n+1}|\mathcal{F}_n]=S_n$$
So it's martingale.
Can I say that finite sequence
$$ \{S_n\}_{i=1}^{n} $$ is martingale or should i have
$$ \{S_n\}_{i=1}^{\infty} $$?
Best Answer
Yes, finite a martingale is a useful thing.
If your particular textbook does not cover the case of a finite martingale $(X_1,X_2,\dots,X_n)$, then extend it by repeating the last value: $(X_1,X_2,\dots,X_n, X_n, X_n, X_n,\dots)$ to get an infinite martingale.
Finite martingales come up, for example, in mathematical finance.