High dimensional integration of kinks and jumps -- smoothing by preintegration
For practitioners in computational finance and high-dimensional integration, this method addresses the bottleneck of non-smooth integrands, but the approach is incremental as it builds on existing preintegration ideas.
The paper introduces a preintegration technique to smooth kinks and jumps in high-dimensional integrands, enabling efficient Quasi-Monte Carlo and Sparse Grid methods. Numerical results on a digital Asian option demonstrate improved efficiency.
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.