0

I am in a Stochastic calc class right now and we are defining the Ito integral. Our definition of a simple process is:

A process $X \in L_2$ is simple if there exists a countable partition $\Pi$ st. $X_t(\omega) = X_{t_j}(\omega)$ for all $t \in [t_j,t_{j+1})$ for all $\omega \in \Omega$.

There is a note immediately after that says:

It is important to note that we assume the partition does not depend on $\omega$. Thus not every piece-wise constant process is a simple process. Give an example of such.

I can not seem to think of an example, any guidance would be appreciated.

user47475
  • 97
  • 2
    Something like $X_t(\omega) = \boldsymbol{1}_{[\omega,\infty)}(t)$ for an appropriate sample space should work. – parsiad Feb 23 '19 at 21:24
  • Can you include your definition of $L_2$? Is this just all indexed sequences of random variables $(X_t)_{t \in [0,\infty)}$ such that $X_t$ has finite second moment for each $t$? – parsiad Feb 23 '19 at 23:14

0 Answers0