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I have two questions on stochastic integration.

(1) Constructing the Ito integral, there is the following remark in Jacod/Shiryaev (page 46, 2nd edition)

One could define the integral process by 4.30 for all $H$ of the form 4.39 but without measurability conditions on $Y$, that is for simple processes that are not predictable. But the extension is essentially possible for

predictableprocesses only.

I will explain what they mean by 4.29 and 4.30, respectively.

By **4.29**:

either $H=Y1_{[0]},\quad Y \text{ is bounded and }\mathcal{F}_{0}\text{ measurable} $

or

$H=Y1_{(r,s]},\quad r<s, Y \text{ is bounded }\mathcal{F}_{r}-\text{ measurable}$

By **4.30**

$H\pmb{\cdot}X_{t}=0$ if $H=Y1_{[0]}$

and

$H\pmb{\cdot}X_{t}=Y(X_{min(s,t)}-X_{min(r,t)})$ if $Y=1_{(r,s]}$

**My Question**: Why do we need predictability? Isn't it possible to extend the integral beyond simple processes, even if there is no predictability? Consider e.g. the Stratonovich integral.
I see the point, that we need this for Ito isometry... . You see, Iam quite confused.

(2) My second question is concerning the stochastic integration in Protter: For an simple predictable process $H$, given by $H_{t}=H_{0}1_{0}(t)+\sum_{i=1}^{n}H_{i}1_{(T_{i},T_{i+1}]}(t)$, with stopping times $T_{\ell}$, he defines $I_{X}(H)=H_{0}X_{0}+\sum_{i=1}^{n}H_{i}(X_{T_{i+1}}-X_{T_{i}})$

**My Question**: Why does Protter replace fixed time points $t_{i}$ by stopping times $T_{i}$?

Diffusions, Markov Processes and Martingalesby Rogers and Williams. $\endgroup$Brownian Motion, Martingales and Stochastic Integrals. $\endgroup$