High dimensional integration of kinks and jumps -- smoothing by preintegration

TL;DR

The paper introduces a preintegration method to smooth out kinks and jumps in high-dimensional integrands, enhancing the efficiency of Quasi Monte Carlo and Sparse Grid methods, especially in option pricing applications.

Contribution

It demonstrates that preintegration reduces integrand irregularities, enabling high-dimensional integration techniques to perform more efficiently in finance models.

Findings

Preintegration smooths integrands with kinks/jumps.

Preintegrated functions belong to suitable Sobolev spaces.

Improved Quasi Monte Carlo efficiency demonstrated on digital Asian option.

Abstract

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.