On the expected uniform error of geometric Brownian motion approximated by the L\'evy-Ciesielski construction

It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the error. In particular, we show for geometric Brownian motion that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the expected uniform error has an upper bound of order $\mathcal{O}(\sqrt{N/2^N})$, or equivalently, $\mathcal{O}(\sqrt{\ln d/d})$. This upper bound matches the known order for the expected uniform error of the standard Brownian motion. We apply the result to an option pricing example

Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $\varepsilon$-error with a cost of $\mathcal{O}(\varepsilon^{-\theta})$ with $\theta < 2$, and in practice even $\theta \approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for non-smooth problems.Comment: 32 page

The Establishment of the GENEQOL Consortium to Investigate the Genetic Disposition of Patient-Reported Quality-of-Life Outcomes

To our knowledge, no comprehensive, interdisciplinary initiatives have been taken to examine the role of genetic variants on patient-reported quality-of-life outcomes. The overall objective of this paper is to describe the establishment of an international and interdisciplinary consortium, the GENEQOL Consortium, which intends to investigate the genetic disposition of patient-reported quality-of-life outcomes. We have identified five primary patient-reported quality-of-life outcomes as initial targets: negative psychological affect, positive psychological affect, self-rated physical health, pain, and fatigue. The first tangible objective of the GENEQOL Consortium is to develop a list of potential biological pathways, genes and genetic variants involved in these quality-of-life outcomes, by reviewing current genetic knowledge. The second objective is to design a research agenda to investigate and validate those genes and genetic variants of patient-reported quality-of-life outcomes, by creating large datasets. During its first meeting, the Consortium has discussed draft summary documents addressing these questions for each patient-reported quality-of-life outcome. A summary of the primary pathways and robust findings of the genetic variants involved is presented here. The research agenda outlines possible research objectives and approaches to examine these and new quality-of-life domains. Intriguing questions arising from this endeavor are discussed. Insight into the genetic versus environmental components of patient-reported quality-of-life outcomes will ultimately allow us to explore new pathways for improving patient care. If we can identify patients who are susceptible to poor quality of life, we will be able to better target specific clinical interventions to enhance their quality of life and treatment outcomes.quality of life, self-rated health, pain, fatigue, genetic disposition, Patient-Reported Quality-of-Life Outcomes