Abstract. It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n = p, p 2 , . . . for all integers p ≥ 2. This paper provides algorithms for the construction of generating vectors which are finitely extensible for n = p, p 2 , . . . for all integers p ≥ 2. The proofs which show that our algorithms yield good extensible rank-1 lattices are based on a sieve principle. Particularly fast algorithms are obtained by using the fast component-by-component construction of Nuyens and Cools. Analogous results are presented for generating vectors with small weighted star discrepancy.

a b s t r a c tWe study the multivariate integration problem R d f (x) ρ(x) dx, with ρ being a product of univariate probability density functions.We assume that f belongs to a weighted tensor-product reproducing kernel Hilbert space of functions whose mixed first derivatives, when multiplied by a weight function ψ, have bounded L 2 -norms. After mapping into the unit cube [0, 1] d , the transformed integrands are typically unbounded or have huge derivatives near the boundary, and thus fail to lie in the usual function space setting where many good results have been established. In our previous work, we have shown that randomly shifted lattice rules can be constructed component-by-component to achieve a worst case error bound of order O(n −1/2 ) in this new function space setting.Using a more clever proof technique together with more restrictive assumptions, in this article we improve the results by proving that a rate of convergence close to the optimal order O(n −1 ) can be achieved with an appropriate choice of parameters for the function space. The implied constants in the big-O bounds can be independent of d under appropriate conditions on the weights of the function space.

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We study the problem of multivariate integration over R d with integrands of the form f (x) d (x) where d is a probability density function. Practical problems of this form occur commonly in statistics and mathematical finance. The necessary step before applying any quasi-Monte Carlo method is to transform the integral into the unit cube [0, 1] d . However, such transformations often result in integrands which are unbounded near the boundary of the cube, and thus most of the existing theory on quasi-Monte Carlo methods cannot be applied. In this paper we assume that f belongs to some weighted tensor product reproducing kernel Hilbert space H d of functions whose mixed first derivatives, when multiplied by a weight function d , are bounded in the L 2 -norm. We prove that good randomly shifted lattice rules can be constructed component by component to achieve a worst case error of order O(n −1/2 ), where the implied constant can be independent of d. We experiment with the Asian option problem using the rules constructed in several variants of the new function space. Our results are as good as those obtained in the anchored Sobolev spaces and they are significantly better than those obtained by the Monte Carlo method.

Monte Carlo methods are used extensively in computational finance to estimate the price of financial derivative options. In this paper we review the use of quasi-Monte Carlo methods to obtain the same accuracy at a much lower computational cost, and focus on these key ingredients: i) the generation of Sobol ′ and lattice points; ii) reduction of effective dimension using the principal component analysis approach at full potential; and iii) randomization by shifting or digital shifting to give an unbiased estimator with a confidence interval.