Hernan Leövey

Andreas Griewank, Frances Y. Kuo, Hernan Leövey et al.

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.

Adrian Ebert, Hernan Leövey, Dirk Nuyens

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assigns a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (2003) proved that the CBC algorithm achieves the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in [Kuo,2003] the error bound of the algorithm can be made independent of the dimension $d$ and we achieve the same optimal order of convergence for the function spaces from [Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to $O(d \, n \log(n))$ operations, where $n$ denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better when the weights decay slower.