We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.

In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.

An extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.

Risk measurements are clearly central to risk management, in particular for banks, (re)insurance companies, and investment funds. The question of the appropriateness of risk measures for evaluating the risk of financial institutions has been heavily debated, especially after the financial crisis of 2008/2009. Another concern for financial institutions is the pro-cyclicality of risk measurements. In this paper, we extend existing work on the pro-cyclicality of the Value-at-Risk to its main competitors, Expected Shortfall, and Expectile: We compare the pro-cyclicality of historical quantile-based risk estimation, taking into account the market state. To characterise the latter, we propose various estimators of the realised volatility. Considering the family of augmented GARCH(p, q) processes (containing well-known GARCH models and iid models, as special cases), we prove that the strength of pro-cyclicality depends on the three factors: the choice of risk measure and its estimators, the realised volatility estimator and the model considered, but, no matter the choices, the pro-cyclicality is always present. We complement this theoretical analysis by performing simulation studies in the iid case and developing a case study on real data.

This article examines the impact of the largest claims reinsurance treaties on loss reserve of the ceding company. The largest claims reinsurance, known as LCR, and ECOMOR reinsurance treaties are considered to be the two most appropriate reinsurance treaties for large or catastrophe claims. Then, it studies the impact of such treaties on loss reserves. Through a simulation study, it shown that, under a more general situation, the LCR treaty can be a more efficient (in some sense, see below) treaty than the ECOMOR treaty for the ceding company.

We propose a generalized Cramér–Lundberg model of the risk theory of non-life insurance and study its ruin probability. Our model is an extension of that of Dubey (1977) to the case of multiple insureds, where the counting process is a mixed Poisson process and the continuously varying premium rate is determined by a Bayesian rule on the number of claims. We use two proofs to show that, for each fixed value of the safety loading, the ruin probability is the same as that of the classical Cramér–Lundberg model and does not depend on either the distribution of the mixing variable of the driving mixed Poisson process or the number of claim contracts.

This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps.

The rich-get-richer rule reinforces actions that have been frequently chosen in the past. What happens to the evolution of individuals’ inclinations to choose an action when agents interact? Interaction tends to homogenize, while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes have recently been considered in many papers, in which the asymptotic behavior is proven to exhibit almost sure synchronization. In this paper we consider models where, even if interaction among agents is present, absence of synchronization may happen because of the choice of an individual nonlinear reinforcement. We show how these systems can naturally be considered as models for coordination games or technological or opinion dynamics.

In this paper, we consider an extended class of univariate and multivariate generalized Pólya processes and study its properties. In the generalized Pólya process considered in [8], each occurrence of an event increases the stochastic intensity of the counting process. In the extended class studied in this paper, on the contrary, it decreases the stochastic intensity of the process, which induces a kind of negative dependence in the increments in the disjoint time intervals. First, we define the extended class of generalized Pólya processes and derive some preliminary results which will be used in the remaining part of the paper. It is seen that the extended class of generalized Pólya processes can be viewed as generalized pure death processes, where the death rate depends on both the state and the time. Based on the preliminary results, the main properties of the multivariate extended generalized Pólya process and meaningful characterizations are obtained. Finally, possible applications to reliability modeling are briefly discussed.

Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.