Why should we study copulas? I am new to the study of copulas and I would like that someone could provide some examples where they are applied, their usefulness and so on. Any help is appreciated. Thanks in advance.
 A: When you fit the distribution to data, you shall quickly realize that it becomes increasingly difficult to get the data when the number of dimensions increases. So, copulas promise to help address this problem.
Consider this problem. You have join distribution $\mathcal F(X_1,X_2,\dots,X_n)$ and want to fit it to the data set with $m$ observations, where 
each row is an observation of your $n$ variables: $$X_1^1,X_2^1,\dots,X_n^1\\X_1^2,X_2^2,\dots,X_n^2\\\dots\\
X_1^m,X_2^m,\dots,X_n^m\\$$
Imagine that you build a grid with $k$ cells along each $n$ axes, i.e. one axis per variable. You'll end up with a volume of $V=k^n$ cells. So, you have $m$ observations to fill the volume $V$ which produces the density $\rho=m/V=mk^{-n}$. As you increase the number of variables the density $\rho$ quickly declines, exponentially, as a matter of fact. 
This makes it very difficult to fit the joint distribution, because your data set dries up so quickly when the number of variables $n$ increase.
In order to have a decent density of data, you need to increase the number of observations exponentially too, by $k^n$.
This is where the copula may help. What copula does is it asks you to fit marginal distributions, then constructs the joint distribution from marginal. The marginal distribution is: $$\mathcal F_1(X_1)\equiv\int_{X_2,\dots,X_n}\mathcal F(\dots)dX_2\dots dX_n$$
To fit the marginal distribution you only need to fill the $k$ cells along the axis of one variable, so the density of data is $\rho_i=m/k$
With the same number of observations $m$ and the same granularity $k$ along the axis, your density of data is much higher: $$\rho_i>>\rho$$ In other words, the same $m$ data points comprise a relatively much richer data set for fitting the marginal distribution.
Summarizing, the promise of copula is to ease requirements to the size of the data set when fitting the joint distributions.
