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In this article we investigate quasi-Monte Carlo (QMC) methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the QMC estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mu¨ck for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands. r This paper is devoted to quasi-Monte Carlo (QMC) techniques for weighted integration problems of the formwhere H denotes an s-dimensional probability distribution with support K ¼ ½a; b ¼ Q s i¼1 ½a ðiÞ ; b ðiÞ DR s and continuous density hðxÞ: Furthermore, f is a function with singularities on the left boundary of K:Numerical integration problems of this form frequently occur in practice, e.g. in the field of computational finance. A typical example is the estimation of the mean of ARTICLE IN PRESS

Summary.In this article, we will first highlight a method proposed by Hlawka and Mück to generate low-discrepancy sequences with an arbitrary distribution H and discuss its shortcomings. As an alternative, we propose an interpolated inversion method that is also shown to generate H-distributed low-discrepancy sequences, in an effort of order O(N log N ).Finally, we will address the issue of integrating functions with a singularity on the boundaries. Sobol' and Owen proved convergence theorems and orders for the uniform distribution, which we will extend to general distributions. Convergence orders will be proved under certain origin-or corner-avoidance conditions, as well as growth conditions on the integrand and the density. Our results prove that also non-uniform quasi-Monte Carlo methods can be well applied to integrands with a polynomial singularity at the integration boundaries.

In this paper a collective risk reserve process of an insurance portfolio characterized by a homogeneous Poisson claim number process, a constant premium flow and independent and identically distributed claims is considered. In the presence of a non-linear dividend barrier strategy and interest on the free reserve we derive equations for the probability of ruin and the expected present value of dividend payments which give rise to several numerical number-theoretic solution techniques. For various claim size distributions and a parabolic barrier numerical tests and comparisons of these techniques are performed. In particular, the efficiency gain obtained by implementing low-discrepancy sequences instead of pseudorandom sequences is investigated.

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