I am working on the numerical integration of an integral of the following functional form:
$$ \int\limits_{R^{G}} F(x_{1},x_{2},\text{…}x_{G})d\text{Φ}_{\text{Σ}}(x_{1},x_{2},\text{…},x_{G}) $$
Here
$$ \text{Φ}_{\text{Σ}}(x_{1},x_{2},\text{…},x_{G}) $$
is the G-dimensional multivariate normal distribution with correlation matrix $\Sigma$ and F is some function of the constituent marginals.
What I am essentially doing is calculating the expectation of a function over a correlated multivariate normal distribution. Practically, G is expected to be equal to or less than 4 and most often just 2 or 3.
Can some one know-how share any of the fundamental references that tackles the issue.My research yielded some information, and it appears that Gaussian quadrature is one of the preferred ways to approach the problem. I am referring to the book Applied Computational
Economics and Finance by Miranda and Fackler for addressing the implementation aspects of the algorithm.
But, I wanted to get some help from the expert community here on if I am on the right track.
Sorry if it is a repeat, however I searched on the site, and was not able to find a question that matches with what I had.
Best Answer
I can refer you to the software by Alan Genz:
http://www.math.wsu.edu/faculty/genz/software/software.html
Sadly in Fortran, but these are probably the best.
At the site there are many functions for calculation of specific expected values, density distributions, cumulative functions etc. For computation of expected value of arbitrary function (user defined) of a multivariate normal distribution you can use function MVNEXG.
Good luck!