Convergence Analysis of the Approximation Problems for Solving Stochastic Vector Variational Inequality Problems

In this paper, we consider stochastic vector variational inequality problems (SVVIPs). Because of the existence of stochastic variable, the SVVIP may have no solutions generally. For solving this problem, we employ the regularized gap function of SVVIP to the loss function and then give a low-risk conditional value-at-risk (CVaR) model. However, this low-risk CVaR model is difficult to solve by the general constraint optimization algorithm. This is because the objective function is nonsmoothing function, and the objective function contains expectation, which is not easy to be computed. By using the sample average approximation technique and smoothing function, we present the corresponding approximation problems of the low-risk CVaR model to deal with these two difficulties related to the low-risk CVaR model. In addition, for the given approximation problems, we prove the convergence results of global optimal solutions and the convergence results of stationary points, respectively. Finally, a numerical experiment is given.

Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

Synthesizing General Electromagnetic Partially Coherent Sources from Random, Correlated Complex Screens

We present a method to generate any genuine electromagnetic partially coherent source (PCS) from correlated, stochastic complex screens. The method described here can be directly implemented on existing spatial-light-modulator-based vector beam generators and can be used in any application which utilizes electromagnetic PCSs. Our method is based on the genuine cross-spectral density matrix criterion. Applying that criterion, we show that stochastic vector field realizations (corresponding to a desired electromagnetic PCS) can be generated by passing correlated Gaussian random numbers through “filters” with space-variant transfer functions. We include step-by-step instructions on how to generate the electromagnetic PCS field realizations. As an example, we simulate the synthesis of a new electromagnetic PCS. Using Monte Carlo analysis, we compute statistical moments from independent optical field realizations and compare those to the corresponding theory. We find that our method produces the desired source—the correct shape, polarization, and coherence properties—within 600 field realizations.

Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Assessing the potential of plains zebra to maintain African horse sickness in the Western Cape Province, South Africa

African horse sickness (AHS) is a disease of equids that results in a non-tariff barrier to the trade of live equids from affected countries. AHS is endemic in South Africa except for a controlled area in the Western Cape Province (WCP) where sporadic outbreaks have occurred in the past 2 decades. There is potential that the presence of zebra populations, thought to be the natural reservoir hosts for AHS, in the WCP could maintain AHS virus circulation in the area and act as a year-round source of infection for horses. However, it remains unclear whether the epidemiology or the ecological conditions present in the WCP would enable persistent circulation of AHS in the local zebra populations.Here we developed a hybrid deterministic-stochastic vector-host compartmental model of AHS transmission in plains zebra (Equus quagga), where host populations are age- and sex-structured and for which population and AHS transmission dynamics are modulated by rainfall and temperature conditions. Using this model, we showed that populations of plains zebra present in the WCP are not sufficiently large for AHS introduction events to become endemic and that coastal populations of zebra need to be >2500 individuals for AHS to persist >2 years, even if zebras are infectious for more than 50 days. AHS cannot become endemic in the coastal population of the WCP unless the zebra population involves at least 50,000 individuals. Finally, inland populations of plains zebra in the WCP may represent a risk for AHS to persist but would require populations of at least 500 zebras or show unrealistic duration of infectiousness for AHS introduction events to become endemic.Our results provide evidence that the risk of AHS persistence from a single introduction event in a given plains zebra population in the WCP is extremely low and it is unlikely to represent a long-term source of infection for local horses.