Let $\overline{S}= (S_{t}^{0},S_{t}^{1})_{t=1,...,T}$ be a process that is adapted to the Filtration $\mathbb{F}=(\mathcal{F_{t}})_{t=1,...,T}$
Determine whether the following processes are previsible or not:
a) $\xi_{t}=1_{\{S_{t}^{1}>S_{t-1}^{1}\}} $
b) $\xi_{1}=1$, and $\xi_{t}=1_{\{S_{t-1}^{1}>S_{t-2}^{1}\}} $ for $t \geq 2$
c) $\xi_{t}=1_{A}1_{\{t>t_{0}\}} $ and $t_{0}\in \{0,...,T\}$ while $A \in \mathcal{F_{t_{0}}}$
d) $\xi_{t}=1_{\{S_{t}^{1}>S_{0}^{1}\}} $
e) $\xi_{1}=1$, and $\xi_{t}=2\xi_{t-1}1_{\{S_{t-1}^{1}<S_{0}^{1}\}} $ for $t \geq 2$
My ideas:
a) not previsible. Consider the function $\xi_{1}=1_{A}$ and $\xi_{2}=1_{A}+ 1_{B}$ where $A$ and $B$ are disjoint, measurable sets. it is clear that $\sigma (\xi_{1})=\{ A^{c}, A, \Omega, \varnothing\}$ but $B \in \sigma (\xi_{2})$
b) $\{ S_{t-1}^{1} > S_{t-2}^{1} \}=\bigcup \limits_{a \in \mathbb Q}(\{ S_{t-1} > a\} \cap \{ a > S_{t-2}\})\in \mathcal{F}_{t-1}$ and hence $1_{\{S_{t-1}^{1}>S_{t-2}^{1}\}} \in \mathcal{F}_{t-1}$
c) I am pretty sure this process is not previsible. I thought about using $t_{0}$ as a stopping time but this does not help prove a non-previsible case.
d) Intuitively, it is clear that this is not previsible. It may be a cheap shot but can simply reduce this from the example I used in (a)?
e) Note that from (b), $\{S_{t-1}^{1}<S_{0}^{1}\}\in \mathcal{F}_{t-1}$ and if we progress inducitively from $\xi_{1}$ and then we let $\xrightarrow{n \to \infty}$ for $\xi_{n}$, we will see that $\xi_{t} \in \mathcal{F}_{t-1}$ as the product of two $\mathcal{F}_{t-1}-$measurable functions.
I look forward to find corrections/help w.r.t. the above five questions.