Let $X$ be a stochastic process on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$ with value in $\mathbb{R}$ equipped with the Borel sigma algebra and the Lebesgue measure. We know that each of the following properties implies the next (definitions reported at the end of the section)
- Predictable
- Optional
- Progressively measurable
- Adapted and measurable
- Adapted
I have found a simple example of a process that is optional but not predictable (see the end of this post) so that the inclusion $1\subset 2$ is evident and of a process that is adapted but not measurable (see the end of this post) so that the inclusion $4\subset 5$ is evident.
To complete the picture I need an example, which is not obvious to me, of a process that is progressively measurable but not optional. Any ideas?
Example of a process optional but not predictable. The process $$ X_t = \begin{cases} 0 & 0\leq t<1\\ \xi & t\geq 1 \end{cases} $$ where $\xi$ is, for example, a binary random variable $\mathbb{P}[\xi=\pm 1]=p\in(0,1)$. If we consider the filtration $\{F_t^{X}\}_{t\geq 0}$ generated by $X$ we have that any $F_t^{X}$-adapted process must be constant in $[0,1)$ because $F_t^X=\{\emptyset,\Omega\}$ for $0\leq t<1$. If the process is, in addition, also left-continuous, it must be constant in $[0,1]$. Accordingly, any predictable process must be constant in $[0,1]$ and so $X$ is not predictable, but it is optional because it is (trivially) $F_t^X$-adapted and right-continuous with left-limit.
Example of a process adapted but not measurable. It is enough to consider a non-measurable Borel set $A\subseteq\mathbb{R}$ and the constant (in $\omega$) process $X_t(\omega)=\mathbb{1}_{A}(t)$. The process is adapted to any filtration (being constant) but trivially it is not measurable: $$ \{(s,\omega)\in [0,\infty)\times\Omega| X_{s}(\omega)=1\}= A\times\Omega\notin\mathcal{B}([0,\infty))\otimes\mathcal{F}, $$
DEFINITIONS
Consider the product space $\Omega\times[0,\infty)$ equipped with the product sigma algebra $\mathcal{F}\otimes\mathcal{B}([0,\infty))$. A measurable stochastic process is any application
$$
X:\Omega\times[0,\infty)\rightarrow\mathbb{R}
$$
measurable with respect to $\mathcal{F}\otimes\mathcal{B}([0,\infty))/\mathcal{B}([-\infty,\infty])$. The predictable sigma-algebra $\mathcal{P}$ is the sigma-algebra on $\Omega\times[0,\infty)$ generated by all the left-continuous and adapted processes. The optional sigma-algebra $\mathcal{O}$ is the sigma-algebra on $\Omega\times[0,\infty)$ generated by all the right-continuous and adapted processes.
A process is said to be predictable if it is $\mathcal{P}$-measurable and optional if it is $\mathcal{O}$-measurable.
A process is said progressively measurable if, for all $t$, it is measurable with respect to the product sigma-algebra $\mathcal{F}_t\otimes\mathcal{B}([0,t])$.
Finally, it is adapted if, for each $t$, the random variable $X_t(\omega):\Omega\rightarrow[-\infty,\infty]$ is $\mathcal{F}_t$-measurable.