The mean of the integral of an adapted process

Question

I am a newbie in the stochastic theory. I read in a lecture note that the mean of an integral of an adapted process is zero. For instance, Let $$ X_{t}=\int_{0}^{t} s \mathrm{~d} B_{s} $$ So, we have$\mathbb{E}\left(X_{t}\right)=0.$

I cannot make an argument on this.

Answer 1

If $f$ is step function it is easy to see that $\int f(s)dB_s$ has mean $0$. [The integral is a finite sum of terms of the type $c(B_v-B_u)$ where $c$ is a constant and $0 \leq u <v$]. By definition of the stochastic integral you can write the given intergal as a limit in mean square (hence also in mean) of integrals of this type. Hence $E\int_0^{t}sdB_s=0$.