There is a vast literature on copula modelling. Using copulas I can describe the joint law of two (and more) random variables $X$ and $Y$, i.e. $F_{X,Y}(x,y)$. Very often in risk management (credit risk, operational risk, insurance) the task is to model a sum $$Z=X+Y$$ and find its distribution $$F_Z(z) = F_{X+Y}(z)$$
I know several approaches that do not directly use copulas (e.g. commons shock models and mixed compound Poisson models) but how can I elegantly combine a copula model and a model for the sum (without Monte Carlo of course - otherwise it would be easy).
Is there some useful Fourier-transform approach? I had the feeling that in the case of Archimedian copulas there could be a chance (looking at this mixture representation as in e.g. in Embrechts, Frey, McNeil). Who has an idea? Are there any papers on this?
Answer
In general setting this is quite a tough problem and it looks like just switching from regular multivariate probability to copulas doesn't make it easier. In general case you need to rely on numerical methods for integration.
There is a nice overview of the problem in Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 September 2009, Part I, Section 5.3, "The Calculation of the Distribution of the Sum of Risks".
If you want to avoid Monte Carlo methods, you can look at a new deterministic method AEP, which was specially designed to tackle this problem. Recursive and FFT methods are compared here.
I haven't seen any work that deals with this problem specifically for Archimedean copulas.
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