Suppose that I have two stochastic processes that can be represented as $$dX_t= udt + \sigma dB_{1,t}$$ $$dY_t= vdt + \nu dB_{2,t}$$ I know by Ito's Lemma that $$d(X_tY_t)=X_tdY_t + Y_tdX_t + dX_tdY_t $$ What I am unsure of is how to interpret terms like $X_tdY_t$ and $dX_tdY_t$. It is always written down that $dB_{i,t}dB_{j,t}=\delta_{ij}dt$ and $dB_tdt=dtdB_t=(dt)^2=0$. So my intuition would tell me that we should always have $$dX_tdY_t=(udt + \sigma dB_{1,t})(vdt + \nu dB_{2,t})=\sigma\nu dB_{1,t}dB_{2,t} $$ Then the term would vanish unless you somehow know that $dB_{1,t}=dB_{2,t}$. I would think that this could happen if $Y_t=g(X_t)$ for some twice differentiable function $g$.
The main part of my confusion is how to interpret $X_tdY_t$ or how to possibly simplify the term into $u,v,\sigma,\nu$. We can write $X_t=X_0+udt+\sigma dB_{1,t}$. I would hope that we could then write \begin{align*} X_tdY_t &= (X_0+udt + \sigma dB_{1,t})(vdt + \nu dB_{2,t})\\ &= (X_0v)dt + (X_0\nu) dB_{2,t} + (\sigma\nu)dB_{1,t}dB_{2,t} \end{align*} At the same time I could also see that I should read $X_tdY_t$ in the following way $$X_tdY_t = (X_tv)dt + (X_t\sigma)dB_{2,t}= \int_0^t X_s(\omega)v(s,\omega)ds + \int_0^t X_s\sigma(s,\omega)dB_{2,s}(\omega) $$ There is just a lot of notation to handle and it has me lost.