Non-synchronous oscillations in four-dimensional nonlinear semelparous Leslie matrix models

In a certain class of semelparous Leslie matrix models, either a positive equilibrium is stable and an invariant set on the boundary of the nonnegative cone is unstable or vice versa generically if the model dimension is two or three. This dynamic dichotomy is expected to be failed in the four-dimensional case. Our study focuses on a semelparous Leslie matrix model with specific nonlinearities and rigorously proves that the dynamic dichotomy does not hold in the four-dimensional case. This result is derived by showing that the four-dimensional semelparous Leslie matrix model can be uniformly persistent with respect to the boundary of the nonnegative cone even if there exists an unstable positive equilibrium. In such a situation, there are no missing age-classes but population oscillation occurs.     

Global dynamics of a special class of nonlinear semelparous Leslie matrix models

ABSTRACT

This paper considers the dynamics of nonlinear semelparous Leslie matrix models. First, a class of semelparous Leslie matrix models is shown to be dynamically consistent with a certain system of Kolmogorov difference equations with cyclic symmetry. Then, the global dynamics of a special class of the latter is fully determined. Combining together, we obtain a special class of semelparous Leslie matrix models which possesses generically either a globally asymptotically stable positive equilibrium or a globally asymptotically stable cycle. The result shows that the periodic behaviour observed in periodical insects can occur as a globally stable phenomenon.     

濒临灭绝蜥蜴种群模型

本以有灭绝危险的佛罗里达丛林蜥蜴为例，给出基于Leslie矩阵的种群保护模型，讨论了非迁移状态及迁移状态下，对丛林蜥蜴生活环境产生影响的因素，然后设计了对它的保护方案，最后在实例中对此方案进行应用和检验，取得了良好的结果。     

Calibrating affine stochastic mortality models using term assurance premiums

In this paper, we focus on the calibration of affine stochastic mortality models using term assurance premiums. We view term assurance contracts as a “swap” in which policyholders exchange cash flows (premiums vs. benefits) with an insurer analogous to a generic interest rate swap or credit default swap. Using a simple bootstrapping procedure, we derive the term structure of mortality rates from a stream of contract quotes with different maturities. This term structure is used to calibrate the parameters of affine stochastic mortality models where the survival probability is expressed in closed form. The Vasicek, Cox-Ingersoll-Ross, and jump-extended Vasicek models are considered for fitting the survival probabilities term structure. An evaluation of the performance of these models is provided with respect to premiums of three Italian insurance companies.     

Estimation of time-varying ARMA models with Markovian changes in regime

In this paper, we consider the estimation of time-varying ARMA models subject to Markovian changes in regime. We give explicit conditions ensuring consistency and asymptotic normality, as well as the limiting covariance matrix, of least squares and quasi-generalized least-squares estimators.     

Improved delay-dependent stability criteria for continuous system with two additive time-varying delay components

This paper investigated the problem of improved delay-dependent stability criteria for continuous system with two additive time-varying delay components. Free weighting matrices and convex combination method are not involved, which achieves much less numbers of linear matrix inequalities (LMIs) and LMIs scalar decision variables. By taking advantage of integral inequality and new Lyapunov–Krasovskii functional, new less conservative delay-dependent stability criterion is derived. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

杂草种子运动的矩阵模型

人类活动(如耕作等)、捕食者活动(如蚂蚁、蚯蚓等)及一些无生命因素(如下雨、冰雪覆盖等),使土壤中的杂草种子在土壤中各个方向上不断运动,运用L eslie矩阵和M arkov链建立了一个具年龄结构的杂草种子在土壤中随时间动态变化的一般矩阵模型.     

Pitfalls and merits of cointegration-based mortality models

In recent years, joint modelling of the mortality of related populations has received a surge of attention. Several of these models employ cointegration techniques to link underlying factors with the aim of producing coherent projections, i.e. projections with non-diverging mortality rates. Often, however, the factors being analysed are not fully identifiable and arbitrary identification constraints are (inadvertently) allowed to influence the analysis thereby compromising its validity. Taking the widely used Lee–Carter model as an example, we point out the limitations and pitfalls of cointegration analysis when applied to semi-identifiable factors. On the other hand, when properly applied cointegration theory offers a rigorous framework for identifying and testing long-run relations between populations. Although widely used as a model building block, cointegration as an inferential tool is often overlooked in mortality analysis. Our aim with this paper is to raise awareness of the inferential strength of cointegration and to identify the time series models and hypotheses most suitable for mortality analysis. The concluding application to UK mortality shows by example the insights that can be obtained from a full cointegration analysis.     

The age pattern of transitory mortality jumps and its impact on the pricing of catastrophic mortality bonds

To value catastrophic mortality bonds, a number of stochastic mortality models with transitory jump effects have been proposed. Rather than modeling the age pattern of jump effects explicitly, most of the existing models assume that the distributions of jump effects and general mortality improvements across ages are identical. Nevertheless, this assumption does not seem to be in line with what we observe from historical data. In this paper, we address this problem by introducing a Lee–Carter variant that captures the age pattern of mortality jumps by a distinct collection of parameters. The model variant is then further generalized to permit the age pattern of jump effects to vary randomly. We illustrate the two proposed models with mortality data from the United States and English and Welsh populations, and use them to value hypothetical mortality bonds with similar specifications to the Atlas IX Capital Class B note that was launched in 2013. It is found that the features we consider have a significant impact on the estimated prices.     

The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.     

误差协方差阵的经验Bayes估计的收敛速度

本文利用密度的混合偏导数的核估计，构造出线性模型中误差协方差阵的逆的经验Ｂａｙｅｓ（ＥＢ）估计，在一定条件下，还证明了ＥＢ估计的收敛速度可任意接近于１，最后，给出了一个实例。     

Optimal harvesting for nonlinear size-structured population dynamics

A harvesting problem is considered for a size structured population model with separable mortality rate and nonlinear fertility rate. We transform the model to a system of equations and show the existence of a unique solution. We also establish the existence of an optimal harvesting rate which maximizes the total harvest in a given time interval.     

Swiss coherent mortality model as a basis for developing longevity de-risking solutions for Swiss pension funds: A practical approach

Pension funds in Switzerland are exposed to longevity risk possibly to a greater extent than in many other developed economies. The ground for this is a dearth of financial products to combat longevity risk, with a lack of buy-in and very limited variety of buy-out solutions available. The solutions that do exist frequently come at a very high price and many pension funds are in deficit on a buy-out basis. From our point of view creating an approach for evaluating the longevity risk faced by each pension fund and integrating it into dynamic risk budgeting strategies will help Swiss pension funds better understand the mechanism behind different longevity de-risking solutions and decide on the most suitable as well as affordable solution for them. To develop capital market solutions for longevity hedging strategies it is crucial that both hedgers (pension funds) as well as solution providers are able to quantify the longevity risk in the framework of a holistic risk management and to develop an adequate pricing approach.In this publication we present our stochastic coherent mortality model developed for Swiss pension funds based on the reference population of fifteen countries and discuss the robustness of the forecasts relative to the sample period used to fit the model, biological reasonableness of the forecasts and other modelling parameters as well as possible impact on results. The model has taken into account past single population modelling techniques and allows flexible age effect to capture the spread behaviour introduced by the target population. The augmented terms for the spread function are chosen based on their forecast accuracy and a coherent behaviour is expected in the long term. The idea behind is fairly simple and yields a design with both transparency and robustness. The model usage is not limited to Switzerland.     

Estimation of the parameters for unstable AR models

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Dynamics of SIR epidemic models with horizontal and vertical transmissions and constant treatment rates

We investigate the dynamics and bifurcations of SIR epidemic model with horizontal and vertical transmissions and constant treatment rates. It is proved that such SIR epidemic model have up to two positive epidemic equilibria and has no positive disease-free equilibria. We find all the ranges of the parameters involved in the model under which the equilibria of the model are positive. By using the qualitative theory of planar systems and the normal form theory, the phase portraits of each equilibria are obtained. We show that the equilibria of the epidemic system can be saddles, stable nodes, stable or unstable focuses, weak centers or cusps. We prove that the system has the Bogdanov-Takens bifurcations, which exhibit saddle-node bifurcations, Hopf bifurcations and homoclinic bifurcations.     

A bifurcation theorem for nonlinear matrix models of population dynamics

ABSTRACT

We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features.