The expectation of an Itô stochastic integral is zero
$$ E[\int_0^t X(s)dB(s)\,]=0 $$
if
$$ \int_0^t E[X^2(s)]ds\,<\infty $$
It is sometimes possible to check this condition directly if the Itô integrand is simple enough but how would you do it if the integrand is the process itself? For example consider the linear SDE
$$ X(t)=X(0) + \int_0^t a ds + \int_0^t b X(s) dW(s) $$
where W(t) is the standard Brownian motion and a, b are constants. How to show that this condition is satisfied for the Itô integral in this process?
Thanks!
Before you can even ask this question, you need to know that there exists a solution to the linear SDE you have written. Without the existence of a solution, the question does not make sense. The standard theorem about the existence of (strong) solutions to SDEs is proved using Picard iteration. As a byproduct of the proof, one obtains an estimate on $E[X^2(s)]$ that can be used to answer your question.
A good reference is Theorem 5.2.9 in Karatzas & Shreve. In the example you gave, the theorem states that a strong solution exists, and for each fixed $t>0$, there exists a constant $C$ that depends only on $t$, $a$, and $b$, such that \[ E[X^2(s)] \le C(1 + E[X^2(0)])e^{Cs}, \quad 0\le s\le t. \]