## Geometric Approximations to Integration Domains and Numerical Algorithms for Distribution Functions

In this talk we discuss the numerical computation of the distribution function of a random variable that is formed by applying a suitable function $g$ to a random vector $X=(X_1,\ldots,X_d)$.

A standard example in financial mathematics and actuarial science is the (weighted) sum of the components of $X$.

The recent literature has seen a new way to tackle these kind of problems by focusing on the decomposition of the associated integration set.

These approaches generally make use of the fact that (hyper-) rectangles can easily be evaluated via the joint distribution function of $X$ and thus aim to find a suitable decomposition of the integration set in terms of (hyper-) rectangles.

A closely related approach can be used to compute the joint distribution function of a multivariate normal distribution.

Finally we give examples related to Expected-Shortfall optimal portfolio selection.