Risk measures, measures for insolvency risk and economical capital allocation.

In the present paper we consider several measures for the risk that is present in an insurance environment. We look for desirable properties for two types of risk measures, the ones reflecting both negative and positive results, and the measures for insolvency risks dealing with aspects of ruin, as well as their relation to the allocation of economic capital to different business lines or to the different subcompanies constituting a financial conglomerate. The main problem for both types of measurements is that the dependence structure that exists between the different units involved is unknown.Dependence; Requirements; Annuities; Risk; Insurance; Risk measure; Measurement; Dependence structure; Structure;

B-series methods are exactly the affine equivariant methods

Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to \emph{sequences of maps}---one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is \emph{affine equivariant}, meaning it respects all affine maps between affine spaces

A simple proof that comonotonic risks have the convex-largest sum.

In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.Risk; Actuarial; Distribution;

On the distribution of cash-flows using Esscher transforms.

In their seminal paper, Gerber and Shiu (1994) introduced the concept of the Esscher transform for option pricing. As examples they considered the shifted Poisson process, the random walk, a shifted gamma process and a shifted inverse Gaussian process to describe the logarithm of the stock price. In the present paper it is shown how upper and lower bounds in convex order can be obtained when we use these types of models to describe the financial stochasticity for a given cash-flow.Cash flow; Pricing; Processes; Models; Model;

The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge-Kutta method cannot preserve volume.Comment: 41 page

The remapped particle-mesh advection scheme

We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the conservation property, the remapped particle-mesh method is computationally efficient and at least as accurate as current semi-Lagrangian methods based on cubic interpolation. We provide results of tests of the method in the plane, results from incorporating the advection method into a semi-Lagrangian method for the rotating shallow-water equations in planar geometry, and results from extending the method to the surface of a sphere

Accelerating radiation computations for dynamical models with targeted machine learning and code optimization

Atmospheric radiation is the main driver of weather and climate, yet due to a complicated absorption spectrum, the precise treatment of radiative transfer in numerical weather and climate models is computationally unfeasible. Radiation parameterizations need to maximize computational efficiency as well as accuracy, and for predicting the future climate many greenhouse gases need to be included. In this work, neural networks (NNs) were developed to replace the gas optics computations in a modern radiation scheme (RTE+RRTMGP) by using carefully constructed models and training data. The NNs, implemented in Fortran and utilizing BLAS for batched inference, are faster by a factor of 1–6, depending on the software and hardware platforms. We combined the accelerated gas optics with a refactored radiative transfer solver, resulting in clear‐sky longwave (shortwave) fluxes being 3.5 (1.8) faster to compute on an Intel platform. The accuracy, evaluated with benchmark line‐by‐line computations across a large range of atmospheric conditions, is very similar to the original scheme with errors in heating rates and top‐of‐atmosphere radiative forcings typically below 0.1 K day−1 and 0.5 W m−2, respectively. These results show that targeted machine learning, code restructuring techniques, and the use of numerical libraries can yield material gains in efficiency while retaining accuracy

Accelerating Radiation Computations for Dynamical Models With Targeted Machine Learning and Code Optimization

Atmospheric radiation is the main driver of weather and climate, yet due to a complicated absorption spectrum, the precise treatment of radiative transfer in numerical weather and climate models is computationally unfeasible. Radiation parameterizations need to maximize computational efficiency as well as accuracy, and for predicting the future climate many greenhouse gases need to be included. In this work, neural networks (NNs) were developed to replace the gas optics computations in a modern radiation scheme (RTE+RRTMGP) by using carefully constructed models and training data. The NNs, implemented in Fortran and utilizing BLAS for batched inference, are faster by a factor of 1&ndash;6, depending on the software and hardware platforms. We combined the accelerated gas optics with a refactored radiative transfer solver, resulting in clear-sky longwave (shortwave) fluxes being 3.5 (1.8) faster to compute on an Intel platform. The accuracy, evaluated with benchmark line-by-line computations across a large range of atmospheric conditions, is very similar to the original scheme with errors in heating rates and top-of-atmosphere radiative forcings typically below 0.1&nbsp;K&nbsp;day&minus;1 and 0.5&nbsp;W&nbsp;m&minus;2, respectively. These results show that targeted machine learning, code restructuring techniques, and the use of numerical libraries can yield material gains in efficiency while retaining accuracy. </div

MEG in the macaque monkey and human: distinguishing cortical fields in space and time.

Magnetoencephalography (MEG) is an increasingly popular non-invasive tool used to record, on a millisecond timescale, the magnetic field changes generated by cortical neural activity. MEG has the advantage, over fMRI for example, that it is a direct measure of neural activity. In the current investigation we used MEG to measure cortical responses to tactile and auditory stimuli in the macaque monkey. We had two aims. First, we sought to determine whether MEG, a technique that may have low spatial accuracy, could be used to distinguish the location and organization of sensory cortical fields in macaque monkeys, a species with a relatively small brain compared to that of the human. Second, we wanted to examine the temporal dynamics of cortical responses in the macaque monkey relative to the human. We recorded MEG data from anesthetized monkeys and, for comparison, from awake humans that were presented with simple tactile and auditory stimuli. Neural source reconstruction of MEG data showed that primary somatosensory and auditory cortex could be differentiated and, further, that separate representations of the digit and lip within somatosensory cortex could be identified in macaque monkeys as well as humans. We compared the latencies of activity from monkey and human data for the three stimulation types and proposed a correspondence between the neural responses of the two species. We thus demonstrate the feasibility of using MEG in the macaque monkey and provide a non-human primate model for examining the relationship between external evoked magnetic fields and their underlying neural sources