I'm having trouble getting my head around the meaning of the stochastic Ito integral. Specifically: the intuitive meaning of "Stochastic Integral" to me is a function that takes a time $t$ and returns a PDF for the integral of a stochastic process on $[0,t]$. Is this indeed what Ito integrals model? If so, I'm confused about why Ito integrals depend on two stochastic processes (you integrate process A with respect to process B). How does the meaning of the integral change when process B changes?
Thanks for your help!