I have trouble getting comfortable with integration with respect to a stochastic process, because I somehow don't see what is happening when you do such an integration. I have a picture in my head for Riemann and Lebesgue integrals but the randomness here somehow makes such an integral kind of obscure for me. There was a post somewhere stating that stochastic integration is "kind of" Riemann-Stieltjes integration but with $\triangle x$ being a random variable, but I still can't wrap my head around that...
I think a good and easy example where such an integral arises naturally would really help me.