  We show how simple kinks and jumps of otherwise smooth integrands over
$\mathbb{R}^d$ can be dealt with by a preliminary integration with respect to a
single well chosen variable. It is assumed that this preintegration, or
conditional sampling, can be carried out with negligible error, which is the
case in particular for option pricing problems. It is proven that under
appropriate conditions the preintegrated function of $d-1$ variables belongs to
appropriate mixed Sobolev spaces, so potentially allowing high efficiency of
Quasi Monte Carlo and Sparse Grid Methods applied to the preintegrated problem.
The efficiency of applying Quasi Monte Carlo to the preintegrated function are
demonstrated on a digital Asian option using the Principal Component Analysis
factorisation of the covariance matrix.

Numerical Analysis