Bivariate Gumbel Distribution - Joint Survival function I need help for this exercise please:
How do I compute the joint Survival function for the bivariate Gumbel distribution:
$f_{xy} = ((1+\delta x)(1+\delta y)-\delta) e^{-x-y-\delta xy}$
According to this paper the result should be $e^{-x-y-\delta xy}$
[with $\theta = \delta$] (see end of the screenshot):

I don't know how a joint survival function is defined, so any help is appreciated!

The joint survival function of a multivariate random variable $\mathbf X = (X_1,X_2, \ldots, X_n)$ is defined to be the probability that all the variables exceed the argument:
$$S_{\mathbf X}(x_1,x_2,\ldots, x_n) = \Pr(X_1\gt x_1,\ X_2\gt x_2,\ \ldots,\ X_n\gt x_n).$$
For the Gumbel density, then (adopting your notation to avoid indices), the value of the survival function at positive arguments $x$ and $y$ is
$$\begin{aligned}
S_{(X,Y)}(x,y) &= \int_y^\infty\int_x^\infty f_{x,y;\theta}(s,t)\,\mathrm ds \mathrm dt\\
&=\int_y^\infty\int_x^\infty ((1+\theta s)(1+\theta t) - \theta)e^{-(s+t+\theta st)}\,\mathrm ds \mathrm dt\\
&=e^{-(x+y+\theta xy)}.
\end{aligned}$$
That's your last formula.
Formula $(1)$ in the question cannot be a survival function because at the origin it evaluates to
$$F(0,0) = 1 - e^{-0} - e^{-0} + e^{-0-0-\theta(0)(0)} = 1-1-1+1 = 0,$$
but obviously $\Pr(X\gt 0,\ Y\gt 0) = 1.$  That's the distribution function, not the survival function.
