On Øksendal's Lemma 3.5

Question

In Øksendal's text, Stochastic Differential Equations, there is a statement of the Ito Isometry for functions $\phi(t,\omega)$ which are bounded and elementary. This is Lemma 3.5 (p. 19).

Naive Question: Where is boundedness used in his proof?

By request, here is the statement of the lemma:

Lemma 3.5: If $\phi(t,\omega)$ is bounded and elementary, then $$ E[(\int_S^T \phi(t,\omega)dB_t(\omega))^2] = E[\int_S^T \phi(t,\omega)^2 dt]\, . $$

Answer 1

You are correct, this issue would be with the right hand side of the isometry.

However as far as I can tell however this assumption does not seem to be necessary, since according to Oksendal

A function $\phi \in \nu$ is elementary if it has the form $$\phi(t,\omega)= \sum_j e_j(\omega) \chi_{[t_j,t_{j+1})}(t).$$

The redundancy comes from the fact that, to be an element in $\nu$ we must have $$E\bigg[\int \limits_S^T \phi(t,\omega)^2 \mathop{dt}\bigg]<\infty.$$ If we take $\Delta t_j=(t_{j+1}-t_j)$ we see that (just as in the proof of Lemma 3.5) $$E\bigg[\int \limits_S^T \phi(t,\omega)^2 \mathop{dt}\bigg] =\sum_j \Delta t_j \int_\Omega e_j^2 \mathop{d\omega} $$

So for each $j$,we have $\int_\Omega e_j^2 \mathop{d\omega}< \infty $.

So each $e_j$ is a.e bounded, and so is $\phi$.