Dimitrios G. Konstantinides

Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails

In this paper we investigate the ruin probability in the classical risk model under a positive constant interest force. We restrict ourselves to the case where the claim size is heavy-tailed, i.e. the equilibrium distribution function (e.d.f.) of the claim size belongs to a wide subclass of the subexponential distributions. Two-sided estimates for the ruin probability are developed by reduction from the classical model without interest force.

Ruin under interest force and subexponential claims: a simple treatment

Abstract A simple proof of the asymptotic formula for the ruin probability of a risk process with a positive constant interest force [derived earlier by Asmussen (Asmussen, S., 1998. The Annals of Applied Probability 8, 354–374)] is given. The proof is based on a formula obtained by Sundt and Teugels (Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22).

A local limit theorem for random walk maxima with heavy tails

For a random walk with negative mean and heavy-tailed increment distribution F, it is well known that under suitable subexponential assumptions, the distribution [pi] of the maximum has a tail [pi](x,[infinity]) which is asymptotically proportional to . We supplement here this by a local result showing that [pi](x,x+z] is asymptotically proportional to zF(x,[infinity]).

Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations

In this paper we consider the stochastic recurrence equation Y t = A t Y t-1 + B t for an i.i.d. sequence of pairs (A t , B t ) of nonnegative random variables, where we assume that B t is regularly varying with index K > 0 and EA k t x) ∼ c 1 nP(Y 1 > x) as x → ∞ holds for some constant c 1 > 0. For K > 1, we also study the large deviation probabilities P(S n - ES n > x). x ≥ x n , for some sequence x n → ∞ whose growth depends on the heaviness of the tail of the distribution of Y 1 . We show that the relation P(S n - ES n > x) ∼ c 2 nP(Y 1 > x) holds uniformly for x ≥ x n and some constant c 2 > 0. Then we apply the large deviation results to derive bounds for the ruin probability ψ(u) = P(sup n≥1 ((S n - ES n ) -μn) > u) for any μ > 0. We show that ψ(u)∼c 3 uP(Y 1 > u)μ -1 (κ - 1) -1 for some constant c 3 > 0. In contrast to the case of i.i.d. regularly varying Y t s, when the above results hold with c 1 = c 2 = c 3 = 1, the constants c 1 , c 2 and c 3 are different from 1.

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

Let us consider a discrete-time insurance risk model with insurance and financial risks, where the insurance net loss within period and the stochastic discount factor over the interval follow a certain dependence structure for each fixed . Under the assumption that the distribution of net insurance loss within one time period is consistently varying-tailed, precise estimates for finite and infinite time ruin probabilities are derived. Furthermore, these estimates are uniform on the time horizon.

Modeling reliability for wireless sensor node coverage in assistive testbeds

Wireless Sensor Networks (WSNs) is a prevailing technology in assistive environments. Assistive environments may include both home and work spaces such as factories, military installations, industrial spaces, and offices. Critical quality-of-service properties of WSN are reliability, availability, and serviceability. This paper focuses on reliability for healthcare applications. Reliable WSN-based monitoring services can prevent accidents, improve the quality of life, and even help with early health diagnosis and treatments. However, because patients/the elderly may have cognitive or other health problems, the reliability is the dominant factor of quality of services of WSN. This paper presents an approach to analyze the reliability of a WSN with the most popular tree structures. The analysis is based on two distribution models, exponential distribution and Weibull distribution. The simulation results also give options to users on the cost vs. reliability issue.

Characterization of tails through hazard rate and convolution closure properties

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.

Precise Large Deviations for Sums of Negatively Dependent Random Variables with Common Long-Tailed Distributions

We investigate the precise large deviations for negatively dependent random variables. We prove general asymptotic relations for both the partial sums S n for the long tailed distributions and the random sums S t for the subexponential distributions, where the N t is an integer counting process. It is found out that the precise large deviations for negatively dependent random variables are insensitive to this kind of dependence. Finally, we present applications on the classical counting processes, Poisson, and renewal.

Coherent Risk Measures Under Dominated Variation

We study the relation between the properties of the coherent risk measures and of the heavy-tailed distributions from radial subsets of random variables. As a result, a new risk measure is introduced for this type of random variable. Under the assumptions of the Lundberg and renewal risk models, the solvency capital in the class of distributions with dominatedly varying tails is calculated. Further, the existence and uniqueness of the solution in the optimisation problem, associated to the minimisation of the risk over a set of financial positions, is investigated. The optimisation results hold on the \(L^{1+\varepsilon }\)-spaces, for any \(\varepsilon \ge 0\), but the uniqueness collapses on \(L^{1}\), the canonical space for the law-invariant coherent risk measures.

Monetary Risk Functionals on Orlicz Spaces Produced by Set-Valued Risk Maps and Random Measures

In this article we study the construction of coherent or convex risk functionals defined either on an Orlicz heart, either on an Orlicz space, with respect to a Young loss function. The Orlicz heart is taken as a subset of \(L^{0}(\varOmega, \mathcal{F}, \mu)\) endowed with the pointwise partial ordering. We define set-valued risk maps related to this partial ordering. We also derive monetary risk functionals both by the class of coherent set-valued risk maps defined on them. We also use random measures related to heavy-tailed distributions in order to define monetary risk functionals on Orlicz spaces, whose properties are also compared to the previous ones.