Let $Ω = {−1, 0, 1},\; \mathbb{P}({−1}) = 1/4,\; \mathbb{P}({0}) = 1/4,\; \mathbb{P}({1}) = 1/2,\; S_0 = 1,\; S_1(ω) = 1+ω$ and $\mathcal{F}$ from $S$ generated filtration. Determine explicitly the set of reachable $K = {(H · S)_1 : H ∈ H}.$
My approach:
$$ (HX)_1=H_1(S_1-S_0) $$ then we have:
$$ (S_1-S_0)(-1)=S_1(-1)-S_0(-1)=0-=-1 $$ $$ (S_1-S_0)(0)=1-1=0 $$ $$ (S_1-S_0)(1)=2-1=1 $$
Therefore,
$$ S_1-S_0=\mathbb{I} $$ on $$\Omega=\{-1,0,1\}$$ $$\Rightarrow (HX)_1=H_1(S_1-S_0)=H_1 $$ on $\Omega$ $H_1$ is a predictable process on the given filtration $\mathcal{F_0}=\{ \varnothing, \Omega \}$, i.e. $H_1 \in \mathcal{F_0}$ and therefore is $K:$
$$ K=\{H_1\in\mathcal{F_0}|H_1 random variable.\} $$
And this the most explicit form I came up with. However, it's obviously supposed to be more specific, since the probability of each event is given.