I am trying to hone my understanding of martingales with respect to their filtrations.
The definition of a martingale is the following:
Let $(\Omega,\mathscr{F},P)$ be a probability space, $\mathbb{F} = \{\mathscr{F}_t,t\geq 0\}$ a filtration contained in $\mathscr{F}$, and $\{Z(t),t\geq 0 \}$ a stochastic process adapted to $\mathbb{F}$. Then $Z$ is a martingale with respect to $\mathbb{F}$ if $\forall t$, $\mathbb{E}[|Z(t)|] < \infty$, and with probability 1, $$ \mathbb{E}[Z(t)|\mathscr{F}_s] = Z(s).$$
I understand that in most cases $\mathbb{F}$ is implicitly understood to be the 'natural filtration' (the smallest $\sigma$-algebra that makes the process measurable), and the filtration is omitted. All alternative filtrations must include the natural one by definition.
In my understanding, the filtration also doesn't change the stochastic process itself, and not even its expectation for a given time i.e. if we would have to different filtrations $\mathbb{F}$ and $\mathbb{F}^{'}$, we still have $ \mathbb{E}[Z(t)|\mathscr{F}_s] = \mathbb{E}[Z(t)|\mathscr{F}^{'}_s] = Z(s)$ by definition.
So when does the filtration really start to matter? Can you give an example of two filtrations for a process $Z(t)$ which, for example, have a different value for $Pr(Z(t)<\alpha|Z(0)=z)$, or for the expectation of some stopping time? Shouldn't these values be the same regardless of filtration because any filtration must include the natural one to make $Z$ measurable? What object/quantity related to the stochastic process might change depending on the filtration?>
