Insurance Risk Theory Research Articles

Computing two actuarial quantities under multilayer dividend strategy with a constant interest rate: Based on Sinc methods

As risk processes with some kind of dividend strategies receive remarkable attention in the financial market, the analysis of risk quantities in the presence of interest (or return) has become an important issue in insurance risk theory, and the corresponding research should be of concern to actuaries. In this paper, we study a perturbed dual risk model with a constant interest rate and a multilayer threshold strategy. We derive respectively the integral differential equations satisfied by the expected present value of the total dividend until ruin and the Laplace transform at the time of ruin. By applying the Sinc method developed in this paper, we derive the approximate solutions of the integral differential equations satisfied by these two risk quantities. Finally, various numerical examples are provided to demonstrate the feasibility of the Sinc method.

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.

Koncepcja ryzyka socjalnego w polskim prawie ubezpieczeń społecznych

Zarys koncepcji ryzyka socjalnego w polskim prawie ubezpieczeń społecznych stworzono przy wykorzystaniu amerykańskiego (ekonomicznego) podejścia do teorii ryzyka ubezpieczeniowego ujmującego to pojęcie w kategoriach niebezpieczeństwa postrzeganego jako przyczyna realnych zdarzeń. W konsekwencji istota ochrony ubezpieczeniowej sprowadza się do odczuwania komfortu psychicznego (poczucia bezpieczeństwa materialnego) zapewnionego przez sam fakt ubezpieczenia. Tym samym za świadczenie wzajemne z tytułu bezwarunkowo opłacanej składki uważa się przejęcie i ponoszenie ryzyka przez ubezpieczyciela, a nie ewentualne wypłacenie świadczenia ubezpieczeniowego. Mając to na uwadze, ryzyko socjalne należy rozpatrywać nie od strony zaspokojenia potrzeb, jakie ono niesie, gdy się ziści (teoria świadczenia pieniężnego), lecz w kontekście świadczenia ubezpieczeniowej gwarancji „na wypadek” zajścia określonych zdarzeń (teoria ponoszenia ryzyka).

Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model

Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim sizes of sample. However, whether the estimators are asymptotically normal or not is unknown. In this paper, we give the details to verify the asymptotic normality of these estimators and present some simulation examples to support our result.

Introducing the non-homogeneous compound-birth process

ABSTRACT In this paper we introduce a novel continuous-time counting stochastic process that we call non-homogeneous compound-birth process. Our motivation is to apply this process in insurance risk theory and queueing theory. To this aim, we study the process in more detail and provide an iterative formula for its transition probabilities. We also deduce the process's probability generating function in a very general particular case, followed by some applications and examples.

Second order tail approximation for the maxima of randomly weighted sums with applications to ruin theory and numerical examples

In this paper the second order asymptotics for the tail probability of the maxima of randomly weighted sums is established under the assumption that the underlying primary random variables have a regularly varying density as x tends to infinity. In doing so, some mild conditions are imposed on the tails of the random weights, and no any assumption is made on the dependence structure between these weights. What is more, an application to insurance risk theory is presented and some numerical examples are given to show the accuracy of the second order results.

Functional limit theorems for a new class of non-stationary shot noise processes

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.

A queueing/inventory and an insurance risk model

AbstractWe study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = ax. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

On the joint analysis of the total discounted payments to policyholders and shareholders: threshold dividend strategy

AbstractIn insurance risk theory, dividend and aggregate claim amount are of great research interest as they represent the insurance company’s payments to its shareholders and policyholders, respectively. Since the analyses of these two quantities are performed separately in the literature, the companion paper by Cheung et al. generalised the Gerber–Shiu expected discounted penalty function by further incorporating the moments of the aggregate discounted claims until ruin and the discounted dividends until ruin. While Cheung et al. considered the compound Poisson model with a dividend barrier in which ruin occurs almost surely, the present paper looks at this generalised Gerber–Shiu function under a threshold dividend strategy where the insurer has a positive survival probability. Because the Gerber–Shiu function is only defined for sample paths leading to ruin, we will additionally study the joint moments of the aggregate discounted claims and the discounted dividends without ruin occurring. Some explicit formulas are derived when the individual claim distribution follows a combination of exponentials. Numerical illustrations involving the correlation between aggregate discounted claims and discounted dividends are given. For the case where ruin occurs, we additionally compute the correlations between the time of ruin and the above two quantities.

Review of statistical actuarial risk modelling

In this paper, we review some results for insurance risk theory. We first introduce a variety of the insurance risk models proposed thus far. Then, we show that the expected discounted penalty function (the so-called Gerber–Shiu function) can describe some risk indicators. Next, the dividend problem is discussed; more precisely, the (approximated) optimal dividend barrier is derived and other extended dividend strategies introduced. In addition, some modified models depending on reinsurance or tax are introduced. Finally, we discuss the statistical estimation of the ruin probability and the Gerber–Shiu function.

ON SOME PROPERTIES OF A CLASS OF MULTIVARIATE ERLANG MIXTURES WITH INSURANCE APPLICATIONS

AbstractWe discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

We consider the spectrally negative Levy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Further, a new expression for the generalized Dickson's formula is provided.

Meromorphic Lévy processes and their fluctuation identities

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119–146, Electron. J. Probab. 13 (2008) 1672–1701], hyper-exponential and generalized hyper-exponential Lévy processes [Quant. Finance 10 (2010) 629–644], Lamperti-stable processes in [J. Appl. Probab. 43 (2006) 967–983, Probab. Math. Statist. 30 (2010) 1–28, Stochastic Process. Appl. 119 (2009) 980–1000, Bull. Sci. Math. 133 (2009) 355–382], Hypergeometric processes in [Ann. Appl. Probab. 20 (2010) 522–564, Ann. Appl. Probab. 21 (2011) 2171–2190, Bernoulli 17 (2011) 34–59], β-processes in [Ann. Appl. Probab. 20 (2010) 1801–1830] and θ-processes in [J. Appl. Probab. 47 (2010) 1023–1033]. In this paper we introduce a new family of Lévy processes, which we call Meromorphic Lévy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener–Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. The specific structure of the M-class Wiener–Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

An improved closed-form solution for the constrained minimization of the root of a quadratic functional

The problem of minimizing the root of a quadratic functional, subject to a system of affine constraints, occurs in investment portfolio selection, insurance risk theory, tomography, and other areas. We provide a solution that improves on the current published solution by being considerably simpler in computational terms. In particular, a succession of partitions and inversions of large matrices is avoided. Our solution method employs the Lagrangian multiplier method and we give two proofs, one of which is based on the solution of a related convex optimization problem. A geometrically intuitive interpretation of the objective function and of the optimization solution is also given.

On a discrete multi-variate phase-type distribution and its applications (abstract only)

In the second part of this paper, we use the discrete MPH-distributions to model multi-variate insurance claim processes in risk analysis, where claims may arrive in batches, the arrivals of different types of batches may be correlated, and the amounts of different types of claims in a batch may be dependent. This provides one natural approach to model the dependencies among claim frequencies as well claim sizes of different types of risks, which is a very important topic in insurance risk theory. Under certain conditions, it is shown that the total amounts of claims accumulated in some random time horizon are discrete MPH random vectors. Matrixrepresentations of the discrete MPH-distributions are constructed explicitly. Efficient computational methods are developed for computing performance measures of the total claims of different types of claim batches and individual types of claims.

On the Probability and the Time of Insurance Ruin

The estimation of ruin probability has been the central topic in insurance risk theory. In this paper we study the asymptotic behavior of the probability of ruin and the probability that ruin occurs before the end of planning years in the compound Poisson model. The exponential bounds for both probabilities are found to be functions of the rate function in traditional large deviation theory.

How to Model Operational Risk, If You Must. Lecture to The Faculty of Actuaries

INTRODUCTIONThe second Lecturer to the Faculty of Actuaries is Professor Paul Embrechts, Professor of Mathematics at the ETH Zurich (Swiss Federal Institute of Technology, Zurich), specialising in actuarial mathematics and mathematical finance. His previous academic positions include ones at the Universities of Leuven, Limburg and London (Imperial College), and he has held visiting appointments at various other universities. He is an elected Fellow of the Institute of Mathematical Statistics, an Honorary Fellow of the Institute of Actuaries, a Corresponding Member of the Italian Institute of Actuaries, Editor of the ASTIN Bulletin, on the Advisory Board of Finance and Stochastics and Associate Editor of numerous scientific journals. He is a member of the Board of the Swiss Association of Actuaries and belongs to various national and international research and academic advisory committees. His areas of specialisation include insurance risk theory, integrated risk management, the interplay between insurance and finance and the modelling of rare events.

Diffusion approximations for the maximum of a perturbed random walk

Consider a random walk S=(Sn: n≥0) that is ‘perturbed’ by a stationary sequence (ξn: n≥0) to produce the process S=(Sn+ξn: n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of Mn=max{Sk+ξk: 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

Diffusion approximations for the maximum of a perturbed random walk

Consider a random walk S=(S n : n≥0) that is ‘perturbed’ by a stationary sequence (ξ n : n≥0) to produce the process S=(S n +ξ n : n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of M n =max{S k +ξ k : 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory

This paper obtains the uniform estimate for maximum of sums of independent and heavy-tailed random variables with nonnegative random weights, which can be arbitrarily dependent of each other. Then the applications to ruin probabilities in a discrete time risk model with dependent stochastic returns are considered.