Describe all martingales that only take values in $\{−1, 0, 1\}=:\Omega$.
In the first instance i would try to find a filtration of
$$P(\Omega)=\{\emptyset,\{0\},\{1\},\{-1\},\{1,-1\},\{1,0\},\{-1,0\},\Omega\}.$$
Candidates are:
$P(\emptyset),P(\{0\}),P(\{1\})\subseteq P(\{0,1\})\subseteq P(\Omega)$
$P(\emptyset),P(\{0\}),P(\{-1\})\subseteq P(\{0,-1\})\subseteq P(\Omega)$
$P(\emptyset),P(\{0\}),P(\{1\})\subseteq P(\{0,1\})\subseteq P(\Omega)$
$P(\emptyset),P(\{1\}),P(\{-1\})\subseteq P(\{1,-1\})\subseteq P(\Omega)$
I'm not really sure how to proceed. Any assistance or thoughts would be much appreciated.
Any martingale taking values in $\lbrace -1,0,1\rbrace$ has the form $X_n = E[X_\infty|\mathcal{F}_n]$ for some $\lbrace -1,0,1\rbrace$-valued random variable $X_\infty$ and some filtration $(\mathcal{F}_n)$. This is because any such martingale is bounded hence by the martingale convergence theorem it converges a.s. and in $L^1$ to some $X_\infty$ and $X_n = E[X_\infty|\mathcal{F}_n]$. ($X_\infty$ necessarily takes values in $\lbrace -1,0,1\rbrace$ by a.s. convergence). Furthermore, since there are finitely many sub-$\sigma$-fields in $\mathcal{P}(\Omega)$, the filtration has to be stationary, i.e. there exists $\mathcal{F} \subset \Omega$ such that $\mathcal{F}_n = \mathcal{F}$ for $n$ large enough.
In conclusion, every martingale taking values in $\lbrace -1,0,1\rbrace$ is stationary, in the sense that there exists a (nonrandom) integer $N \in \mathbb{N}$ such that $$X_n = E[X_\infty|\mathcal{F}_n], \quad \forall n \leqslant N,$$ and $$X_n = X_\infty, \quad \forall n >N,$$ where $X_\infty$ is some $\lbrace -1,0,1\rbrace$-valued random variable and $\mathcal{F}_1, \ldots, \mathcal{F}_N$ is a (finite) filtration.