Showing that the elementary processes generate the predictable sigma-algebra

Question

We work with respect to a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},P)$

An elementary process is of the form

\begin{equation} \label{eq:1} \xi(t) = Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_k<t\le t_k\}} \end{equation} for ${n\ge 0}$, times ${s_k<t_k}$, ${\mathcal{F}_0}$-measurable random variable ${Z_0}$ and ${\mathcal{F}_{s_k}}$-measurable random variables ${Z_k}$.

The predictable sigma-algebra on ${{\mathbb R}^+\times\Omega}$ is the sigma-algebra generated by the left-continuous and adapted processes.

How can we show that the elementary processes also generate the predictable sigma-algebra ?

Answer 1

Denote the sigma-algebra generated by predictable processes by $\mathcal{E}$ and the sigma-algebra generated by left-continuous and adapted processes by $\mathcal{H}$. We want to show that $\mathcal{E} = \mathcal{H}$.

Clearly the elementary processes $$ \xi(t) = Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_k<t\le t_k\}} $$ are left-continuous and adapted (and also constant on the intervals $(t_k,t_{k+1})$). Thus at leas we have $\mathcal{E} \subset \mathcal{H}$ i.e. every predictable process is left-continuous and adapted.

Conversely if $X$ is adapted and left-continuous process it is a limit of piecewise constant functions. More presicely $X$ is the limit (as $n \to \infty$) of $$ X_t = X_01_{\{t=0\}}+\sum_{i=1}^\infty X_{\{(i-1)/n\}} 1_{\{(i-1)/n < t \leq i/n\}}. $$ Thus also $\mathcal{H} \subset \mathcal{E}$.