It seems to me there are two notions for random variables / processes which get labeled square-integrable:
- $EX^2_t<\infty \; \forall t$
- $E \int^t_0 X_s^2 \; ds < \infty \; \forall t$
I suppose (1) is that $X_t$ is a square-integrable variable for all $t$ and (2) is square-integrable in both variables, but only "progressively" in $t$
My questions: What is the relationship between these two conditions? What purpose do they serve and in what context are they likely to be used?
Thank you.
By Fubini's Theorem, (2) is equivalent to
$$\int_0^t E[X_s^2] ds < \infty$$
If $$E[X_s^2] = \infty$$ then $$\int_0^t E[X_s^2] ds = \infty \ ↯$$
Hence $$E[X_s^2] < \infty$$
Some specifics with $s$ and $t$ but I think (2) implies (1)