Simple example of an adapted process that is a martingale for one filtration but not for another

Question

I have seen this question. I haven't had Brownian motion yet, is there a simple, more basic example?

Answer 1

Here's a problem which has a simple solution that answers your question:

Suppose $(M_n)$ is a martingale with respect to a filtration $(\mathcal{F}_n)$. Then $\mathcal{F}_n \subset \mathcal{F}_{n+1}$, so $M_n$ is definitely $\mathcal{F}_{n+1}$-measurable.

Define $\mathcal{G}_n = \mathcal{F}_{n+1}$. When is $(M_n)$ a martingale with respect to the filtration $(\mathcal{G}_n)$?

Here's the solution in case you need it:

If $(M_n)$ is a martingale with respect to $(\mathcal{G}_n)$, then $$M_n = \mathbb{E}[M_{n+1}\mid \mathcal{G}_n] = \mathbb{E}[M_{n+1}\mid \mathcal{F}_{n+1}] = M_{n+1}$$

This resolves your question because

there exist non-constant martingales indexed by $\mathbb{N}$. E.g. A simple random walk on $\mathbb{Z}$