Expectation of Ito integral, part 2, and Fubini theorem

Question

I previously asked a question (Expectation of Ito integral). I have additional questions on the same subject.

Let's say that we have an Ito process such as

$$ X(t)=X(0) + \int_0^t a ds + \int_0^t b X(s) dW(s) $$

where a and b are constants and W(t) is the standard Brownian motion.

Using Itô's formula for $X^2$, we have

$$ X(t)^2=X(0)^2 + \int_0^t (2 a X(s) + b^2 X(s)^2 )ds + 2ab\int_0^t X(s)^2 dW(s) $$

In my calculations I need the fact that $E[\int_0^t X(s)^2 dW(s)]=0$, which requires that $E[\int_0^t |X(s)|^4 s]<\infty$. We know that $E[\int_0^t |X(s)|^2 s]<\infty$ from the existence of the strong solution for the first SDE. Is there a link between them?

The second question is the justification of the application of Fubini as follows

$$ E[\int_0^t (2 a X(s) + b^2 X(s)^2 )ds] = \int_0^t (2 a E[X(s)] + b^2 E[X(s)^2] )ds $$

Fubini can be applied if $E[\int_0^t |X(s)|ds]<\infty$ and $E[\int_0^t |X(s)^2|ds]<\infty$. This very much resembles the first condition, however it is not entirely clear how I can tie them up (for both $X$ and $X^2$ processes). Perhaps Jensen's inequality can be used $E(X)^2\leq E(X^2)$ (but I can only use it for a partition within the integral)?

Thanks and kind regards!

Answer 1

In my answer to your previous question, I mentioned Theorem 5.2.9 in Karatzas & Shreve. The proof of that theorem, part of which is relegated to Problem 5.2.10 (which has a worked solution), can be modified to yield an improved result on moment estimates. It is mostly just an exercise in using the Burkholder-Davis-Gundy inequalities, but I have worked out the details and posted them to the "Lecture notes" section of my website. Here is a direct link to the short note. In the note, you will want to use (4) with $m=2$ and $m=1$, and Corollary 3 with $r=1$. The end result is that in your example, everything will work out as long as $E|X(0)|^4<\infty$.