Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution for $X$ and $Y$.
Does this mean that $\int_{0}^{t}g(s)dX_{s}$ and $\int_{0}^{t}g(s)dY_{s}$ have the same bivariate distribution? So again in the case of regularly correlated BM's, this would imply a bivariate normal distribution but with different mean, and covariance matrices?
Does this work for all distributions? Lets say $X_{t}$ and $Y_{t}$ are distributed with copula $C$, does this mean that $\int_{0}^{t}g(s)dX_{s}$ and $\int_{0}^{t}g(s)dY_{s}$ are also distributed with copula $C$?
And if so, why?