Brownian bridge and principal component analysis: towards removing the curse of dimensionality

Abstract

High-dimensional integrals occur in a variety of areas, including mathematical finance. In the classical settings, multivariate integration problems suffer from the curse of dimensionality. To vanquish the curse of dimensionality, one may shrink the function class. Here, we use weighted function spaces, in which groups of variables are associated with weights, in order to capture the different importance of each group of variables. For practical applications, the principal difficulty is then in choosing the ‘right’ weights for a given problem or a class of problems. We work in weighted reproducing kernel Hilbert spaces (with ‘general’ rather than ‘product’ weights). We first present a principle to find the good weights for a given problem. This general approach is then applied to a simplified high-dimensional problem from finance, in which the dimension is the number of discrete timesteps for a price of a risky asset which follows geometric Brownian motion. The second focus of this paper is on the dimension reduction techniques, such as the Brownian bridge (BB) and the principal component analysis (PCA). It turns out that the behaviour of the model problem is dramatically improved when quasi-Monte Carlo method is used in conjunction with the dimension reduction techniques: if the right weights are used in every case, then the integration error can be bounded independently of the dimension, whereas without BB or PCA the error bound depends exponentially on the dimension. Finally, for this model problem we show how to construct shifted lattice rules which, when used in conjunction with BB or PCA, yield integration errors converging as O(n−3/4+δ) or O(n−1+δ) (for arbitrary small δ > 0), respectively, independently of the dimension. Thus, in both cases well-designed algorithms can avoid the curse of dimensionality, with PCA having an advantage over BB with respect to the proved order of convergence.