I am trying to work through protters book and some supplementary notes on my own, but am stuck. I wish to know when a simple predictable process is adapted.
Our definition of simple predictable will be: $X$ is simple predictable if there exist bounded random variables $Z_k$, $k = 0,1,\cdots,n$ and finite stopping times $0 = T_1 \leq T_2 \leq \cdots \leq T_n < \infty $ such that
$X = Z_0\mathbf{1}_{\{0\}} + \sum_{k = 1}^{n-1} Z_k\mathbf{1}_{(T_k,T_{k+1}]}$ and $Z_k$ is $\mathcal{F}_{T_{k}}$ measurable.
Since predictable processes in general are just those processes that are measurable with respect to the predictable sigma algebra generated by left continuous and adapted processes, predictable processes must therefore be adapted. However, starting with the definition above, I find it hard to see that the simple predictable processes are adapted.
I would like to show that these processes generate the predictible processes. I already know that processes of the form $\mathbf{1}_{(S,T]}$ are left continuous and adapted… adapted since for any $t \geq 0$, the function $h_t(\omega) = \mathbf{1}_{(S,T]}(t,\omega)= \mathbf{1}_{\{S<t\} \cap \{T \geq t\}}(\omega)$ is $\mathcal{F_t}$ measurable if we assume that our filtration is right continuous.
I would like to know why something of the form $Z_k\mathbf{1}_{(T_k,T_{k+1}]}$ is adapted if $Z_k$ is $\mathcal{F}_{T_k}$ measurable like assumed in protter.
Thanks
Update:
For a simple example, let $S \leq T$ be finite stopping times, $Z$ a bounded $\mathcal{F}_S$ measurable random variable and $t \geq 0$. Put $g_t(\omega) = Z(\omega)\mathbf{1}_{(S,T]}(t,\omega)$. We want to show that $g_t$ is $\mathcal{F}_t$ measurable.
I understand that if $0 \not \in B$, then $g_t^{-1}(B) = (Z^{-1}(B) \cap \{S \leq t\}) \cap \{S = t\}^c \cap \{T \geq t\} \in \mathcal{F}_t$.
But I cannot figure out why $g_t^{-1}(0) = \{Z = 0\} \cup \{S \geq t\} \cup \{T < t\}$ must be in $\mathcal{F_t}$.
Best Answer
It seems to me that any process called predictable should be adapted, but who knows, maybe someone somewhere has defined predictable but not adapted processes?
In any case, I think your $X$ is adapted for the following reason. The question is whether $\{X_t\in B\}\in\mathcal F_t$ for all Borel sets $B$ and $t> 0$. And we can rewrite $\{X_t\in B\}$ as \begin{align} \{X_t\in B\} & = \cup_{k=1}^n(\{Z_k\in B\} \cap \{t\in(T_k,T_{k+1}]\} )\\ & = \cup_{k=1}^n(\{Z_k\in B\} \cap \{T_k\le t\} \cap \{t=T_k\}^c \cap \{t\le T_{k+1}\} ) \end{align} where I take $T_{n+1}$ to be $\infty$ for convenience. Now, $\{Z_k\in B\} \cap \{T_k\le t\}\in\mathcal F_t$ by the assumption $Z_k\in\mathcal F_{T_k}$ and so are $\{t=T_k\}^c$ and $\{t\le T_{k+1}\}$ as they are stopping times. The conclusion follows.