The bivariate gaussian copula is defined as
$$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy$$
where $\Phi$ is the cumulative function of a standard gaussian and $\rho$ is a parameter in [-1,1].
It seems that the bivariate gaussian copula family is positively ordered i.e.
$$ ρ_1<ρ_2⇒C_{ρ_1}(u,v) < C_{ρ_2}(u,v)$$
I don't know how to prove this result which seems quite popular and is useful in mathematical finance. The result is mentioned in Copulas for Finance.
I have tried to prove that $\frac{\partial f}{\partial \rho} > 0$ where $f(\rho,x,y)$ is defined as
$$f(\rho,x,y) = \frac{1}{2π\sqrt{1-ρ^2}}×exp(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})$$
However I could not conclude. I end up with
$$\frac{\partial f}{\partial \rho} = \frac{exp(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})}{2π\sqrt{1-ρ^2}}×\frac{\rho(1-\rho^2) + xy(1+\rho^2)-\rho(x^2+y^2)}{(1-\rho^2)^2}$$
Given the expression above, I don't think that $\frac{\partial f}{\partial \rho} > 0$. There must be another way to prove this result.
Any ideas? Any references (articles, books) with the proof would also be greatly appreciated.