In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first nontrivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or nonlinear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter, J. Complexity, 26 (2010), pp. 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of highdimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group.We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.

The paper demonstrates that a pure-diffusion 3/2 model is able to capture the observed upward-sloping implied volatility skew in VIX options. This observation contradicts a common perception in the literature that jumps are required for the consistent modelling of equity and VIX derivatives. The pure-diffusion model, however, struggles to reproduce the smile in the implied volatilities of short-term index options. The pronounced implied volatility smile produces artificially inflated fitted parameters, resulting in unrealistically high VIX option implied volatilities. To remedy these shortcomings, jumps are introduced. The resulting model is able to better fit short-term index option implied volatilities while producing more realistic VIX option implied volatilities, without a loss in tractability.

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a b s t r a c tHigher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of ''good'' quality has recently been established, but their construction was not addressed.We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time -under certain conditions on the weights -(strong) polynomial tractability. Combining this approach with a sievetype algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.