A Continuous Time GARCH Process Driven by a Lévy Process: Stationarity and Second Order Behaviour

We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, tosuggest an extension of the (G)ARCH concept to continuous time processes. Our "COGARCH" (continuous time GARCH) model, based on a single background driving Levy process, is different from, though related to, other continuous time stochastic volatility models that have been proposed. The model generalises the essential features of discrete time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties

Stationarity and second order behaviour of discrete and continuous time GARCH(1,1) processes

We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. The models, based on a single background driving Levy process, are different from, though related to, other continuous time stochastic volatility models that have been proposed. Our models generalise the essential features of discrete time GARCH processes, and are amenable to further analysis, possessing useful Markovian and stationarity properties

Arc Statistics in Clusters: Galaxy Contribution

The frequency with which background galaxies appear as long arcs as a result of gravitational lensing by foreground clusters of galaxies has recently been found to be a very sensitive probe of cosmological models by Bartelmann et al. (1998). They have found that such arcs would be expected far less frequently than observed (by an order of magnitude) in the currently favored model for the universe, with a large cosmological constant $\Omega_\Lambda \sim 0.7$. Here we analyze whether including the effect of cluster galaxies on the likelihood of clusters to generate long-arc images of background galaxies can change the statistics. Taking into account a variety of constraints on the properties of cluster galaxies, we find that there are not enough sufficiently massive galaxies in a cluster for them to significantly enhance the cross section of clusters to generate long arcs. We find that cluster galaxies typically enhance the cross section by only $\lesssim 15$.Comment: 19 pages, 1 figure, uses aasms4.sty, submitted to Ap

Breaking the Disk/Halo Degeneracy with Gravitational Lensing

The degeneracy between the disk and the dark matter contribution to galaxy rotation curves remains an important uncertainty in our understanding of disk galaxies. Here we discuss a new method for breaking this degeneracy using gravitational lensing by spiral galaxies, and apply this method to the spiral lens B1600+434 as an example. The combined image and lens photometry constraints allow models for B1600+434 with either a nearly singular dark matter halo, or a halo with a sizable core. A maximum disk model is ruled out with high confidence. Further information, such as the circular velocity of this galaxy, will help break the degeneracies. Future studies of spiral galaxy lenses will be able to determine the relative contribution of disk, bulge, and halo to the mass in the inner parts of galaxies.Comment: Replaced with minor revisions, a typo fixed, and reference added; 21 pages, 8 figures, ApJ accepte

Altered hippocampal function in major depression despite intact structure and resting perfusion

Background: Hippocampal volume reductions in major depression have been frequently reported. However, evidence for functional abnormalities in the same region in depression has been less clear. We investigated hippocampal function in depression using functional magnetic resonance imaging (fMRI) and neuropsychological tasks tapping spatial memory function, with complementing measures of hippocampal volume and resting blood flow to aid interpretation. Method: A total of 20 patients with major depressive disorder (MDD) and a matched group of 20 healthy individuals participated. Participants underwent multimodal magnetic resonance imaging (MRI): fMRI during a spatial memory task, and structural MRI and resting blood flow measurements of the hippocampal region using arterial spin labelling. An offline battery of neuropsychological tests, including several measures of spatial memory, was also completed. Results: The fMRI analysis showed significant group differences in bilateral anterior regions of the hippocampus. While control participants showed task-dependent differences in blood oxygen level-dependent (BOLD) signal, depressed patients did not. No group differences were detected with regard to hippocampal volume or resting blood flow. Patients showed reduced performance in several offline neuropsychological measures. All group differences were independent of differences in hippocampal volume and hippocampal blood flow. Conclusions: Functional abnormalities of the hippocampus can be observed in patients with MDD even when the volume and resting perfusion in the same region appear normal. This suggests that changes in hippocampal function can be observed independently of structural abnormalities of the hippocampus in depression

Convergence to Stable Limits for Ratios of Trimmed Lévy Processes and their Jumps

We derive characteristic function identities for conditional distributions of an $r$-trimmed L\'evy process given its $r$ largest jumps up to a designated time $t$. Assuming the underlying L\'evy process is in the domain of attraction of a stable process as t\dto 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in \cite{KeveiMason2014} and produces new discrete distributions on the infinite simplex in the limit

NICMOS images of JVAS/CLASS gravitational lens systems

We present Hubble Space Telescope (HST) infrared images of four gravitational lens systems from the JVAS/CLASS gravitational lens survey and compare the new infrared HST pictures with previously published WFPC2 HST optical images and radio maps. Apart from the wealth of information that we get from the flux ratios and accurate positions and separations of the components of the lens systems that we can use as inputs for better constraints on the lens models we are able to discriminate between reddening and optical/radio microlensing as the possible cause of differences observed in the flux ratios of the components across the three wavelength bands. Substantial reddening has been known to be present in the lens system B1600+434 and has been further confirmed by the present infrared data. In the two systems B0712+472 and B1030+074 microlensing has been pinpointed down as the main cause of the flux ratio discrepancy both in the optical/infrared and in the radio, the radio possibly caused by the substructure revealed in the lensing galaxies. In B0218+357 however the results are still not conclusive. If we are actually seeing the two "true" components of the lens system then the flux ratio differences are attributed to a combination of microlensing and reddening or alternatively due to some variability in at least one of the images. Otherwise the second "true" component of B0218+357 maybe completely absorbed by a molecular cloud and the anomalous flux density ratios and large difference in separation between the optical/infrared and radio that we see can be explained by emission from either a foreground object or from part of the lensing galaxy.Comment: 10 pages, 4 figures (original higher resolution figures can be obtained at the e-mail above), to appear in MNRAS (accepted