On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results

Multilevel Monte Carlo methods

The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research

Host preferences and differential contributions of deciduous tree species shape mycorrhizal species richness in a mixed Central European forest

Mycorrhizal species richness and host ranges were investigated in mixed deciduous stands composed of Fagus sylvatica, Tilia spp., Carpinus betulus, Acer spp., and Fraxinus excelsior. Acer and Fraxinus were colonized by arbuscular mycorrhizas and contributed 5% to total stand mycorrhizal fungal species richness. Tilia hosted similar and Carpinus half the number of ectomycorrhizal (EM) fungal taxa compared with Fagus (75 putative taxa). The relative abundance of the host tree the EM fungal richness decreased in the order Fagus > Tilia >> Carpinus. After correction for similar sampling intensities, EM fungal species richness of Carpinus was still about 30–40% lower than that of Fagus and Tilia. About 10% of the mycorrhizal species were shared among the EM forming trees; 29% were associated with two host tree species and 61% with only one of the hosts. The latter group consisted mainly of rare EM fungal species colonizing about 20% of the root tips and included known specialists but also putative non-host associations such as conifer or shrub mycorrhizas. Our data indicate that EM fungal species richness was associated with tree identity and suggest that Fagus secures EM fungal diversity in an ecosystem since it shared more common EM fungi with Tilia and Carpinus than the latter two among each other

A Dual Fluorescence–Spin Label Probe for Visualization and Quantification of Target Molecules in Tissue by Multiplexed FLIM–EPR Spectroscopy

Simultaneous visualization and concentration quantification of molecules in biological tissue is an important though challenging goal. The advantages of fluorescence lifetime imaging microscopy (FLIM) for visualization, and electron paramagnetic resonance (EPR) spectroscopy for quantification are complementary. Their combination in a multiplexed approach promises a successful but ambitious strategy because of spin label-mediated fluorescence quenching. Here, we solved this problem and present the molecular design of a dual label (DL) compound comprising a highly fluorescent dye together with an EPR spin probe, which also renders the fluorescence lifetime to be concentration sensitive. The DL can easily be coupled to the biomolecule of choice, enabling in vivo and in vitro applications. This novel approach paves the way for elegant studies ranging from fundamental biological investigations to preclinical drug research, as shown in proof-of-principle penetration experiments in human skin ex vivo