Gunther Leobacher

Jan Baldeaux, Josef Dick, Gunther Leobacher et al.

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 high-dimensional 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.

Josef Dick, Christian Irrgeher, Gunther Leobacher et al.

We study the numerical approximation of integrals over $\mathbb{R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via $L_2$ norms of the derivatives of the function. We map higher order digital nets from the unit cube to a suitable subcube of $\mathbb{R}^s$ via a linear transformation and show that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.

Gunther Leobacher, Michaela Szölgyenyi

We prove strong convergence of order $1/4-ε$ for arbitrarily small $ε>0$ of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

Gunther Leobacher, Michaela Szölgyenyi

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.

Christian Irrgeher, Gunther Leobacher

It has been found empirically that quasi-Monte Carlo methods are often efficient for very high-dimensional problems, that is, with dimension in the hundreds or even thousands. The common explanation for this surprising fact is that those functions for which this holds true behave rather like low-dimensional functions in that only few of the coordinates have a size- able influence on its value. However, this statement may be true only after applying a suitable orthogonal transform to the input data, like utilizing the Brownian bridge construction or principal component analysis construction. We study the effect of general orthogonal transforms on functions on $\mathbb{R}^d$ which are ele- ments of certain weighted reproducing kernel Hilbert spaces. The notion of Hermite spaces is defined and it is shown that there are examples which admit tractability of integration. We translate the action of the orthogonal transform of $\mathbb{R}^d$ into an action on the Hermite coefficients and we give examples where orthogonal transforms have a dramatic effect on the weighted norm, thus providing an explanation for the efficiency of using suitable orthogonal transforms.

Gunther Leobacher, Michaela Szölgyenyi

For time-homogeneous stochastic differential equations (SDEs) it is enough to know that the coefficients are Lipschitz to conclude existence and uniqueness of a solution, as well as the existence of a strongly convergent numerical method for its approximation. Here we introduce a notion of piecewise Lipschitz functions and study SDEs with a drift coefficient satisfying only this weaker regularity condition. For these SDEs we can construct a strongly convergent approximation scheme, if the set of discontinuities is a sufficiently smooth hypersurface satisfying the geometrical property of being of positive reach. We then arrive at similar conclusions as in the Lipschitz case. We will see that, although SDEs are in the center of our interest, we will talk surprisingly little about probability theory here.

Gunther Leobacher

One of the main practical applications of quasi-Monte Carlo (QMC) methods is the valuation of financial derivatives. We aim to give a short introduction into option pricing and show how it is facilitated using QMC. We give some practical examples for illustration.

Christian Irrgeher, Gunther Leobacher

There are a number of situations where, when computing prices of financial derivatives using quasi-Monte Carlo (QMC), it turns out to be beneficial to apply an orthogonal transform to the standard normal input variables. Sometimes those transforms can be computed in time $O(n\log(n))$ for problems depending on $n$ input variables. Among those are classical methods like the Brownian bridge construction and principal component analysis (PCA) construction for Brownian paths. Building on preliminary work by Imai and Tan [3] as well as Wang and Sloan [13], where the authors try to find optimal orthogonal transform for given problems, we present how those transforms can be approximated by others that are fast to compute. We further present a new regression-based method for finding a Householder reflection which turns out to be very efficient for a wide range of problems. We apply these methods to several very high-dimensional examples from finance.

Christian Irrgeher, Gunther Leobacher

We combine a generic method for finding fast orthogonal transforms for a given quasi-Monte Carlo integration problem with the multilevel Monte Carlo method. It is shown by example that this combined method can vastly improve the efficiency of quasi-Monte Carlo.

Josef Dick, Peter Kritzer, Gunther Leobacher et al.

QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $α$. Using Parseval's identity, this smoothness can be defined for all real numbers $α> 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $α$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-μ(1-λ)/2})$ for any $1/μ<λ\le 1$, where $μ>1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(γ_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty γ_j^τ< \infty$ for some $1/(2α) < τ\le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $τ$ is close to $1/(2 α)$, namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty γ_j \max\{1, \log γ_j^{-1}\}^{μ(1-λ)} < \infty$ for some $1/μ< λ\le 1$. The essential tool in our analysis is a $\log$-scale Jensen's inequality. The results described above also apply to integration in $\log$-cosine spaces using tent-transformed lattice rules.