On stochastic integration

Question

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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 predictable processes 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 $T_{i}$?

Answer 1

Jacod/Shiryaev mention in the next paragraph "a fundamental result by Bichteler, Dellacherie and Mokobodzki, which explains why the space of semimartingales is so important.". The Bichteler-Dellacherie theorem and here say that for semimartingales this the most general construction possible.

Theorem 1 (Bichteler-Dellacherie) For a cadlag adapted process X, the following are equivalent.

• X is a semimartingale.

• For each ${t\geq 0}$, the set given by $$\displaystyle \left\{\int_0^t\xi\,dX\colon\xi{\rm\ is~ simple~ predictable },\ \vert\xi\vert\le1\right\} $$ is bounded in probability.

• X is the sum of a local martingale and an FV process.

The Stratonovich integral is defined in terms of the Itô integral How can we define the Stratonovich integral rigorously? where we indeed need to use predictability in order to use martingale tools.

The generalization to increasing stopping times is quite interesting because indeed all that is needed is adaptability. One can imagine situations in numerical analysis that prefers to work with stopping times as in adapted grids or random environment.