Large deviation principles in boundary problems for compound renewal processes

We find explicit logarithmic asymptotics for the probability of events related to the intersection (or nonintersection) of arbitrary remote boundaries by the trajectory of a compound renewal process.     

Mathematical Study of an Inflammatory Model for Atherosclerosis: A Nonlinear Renewal Equation

In this work we study the coupling of a nonlinear renewal equation to an ordinary differential equation. We start with existence and uniqueness issues for the coupled equations and, in particular cases, we study the long-time behaviour. The novelty here is the nonlinearity in the renewal equation. This model arises in the context of atherosclerosis. The renewal part accounts for the inflammatory process: leucocyte recruitment in the arterial wall, differentiation when internalizing low-density lipoprotein (LDL) and death. The ordinary differential equation describes the LDL dynamics in the arterial wall, leucocyte absorption and release in the blood.

     

The impact of data collection on fill rate performance in the (R,s, Q) inventory model

In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.     

The Gerber-Shiu function and the generalized Cramér-Lundberg model

In this paper we extend some results in Cramér [7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang(n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.     

Exit problems for the difference of a compound Poisson process and a compound renewal process

In this paper we solve a two-sided exit problem for a difference of a compound Poisson process and a compound renewal process. More specifically, we determine the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this time instant. The results obtained are applied to solve the two-sided exit problem for a particular class of stochastic processes, i.e. the difference of the compound Poisson process and the renewal process whose jumps are exponentially distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. We determine the Laplace transforms of the busy period of the systems M ^? |G ^δ |1|B, G ^δ |M ^? |1|B in case when δ～exp?(λ). Additionally, we prove the weak convergence of the two-boundary characteristics of the process to the corresponding functionals of the standard Wiener process.     

On ruin probabilities with risky investments in a stock with stochastic volatility

We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities.

     

The Availability and Hazard of a System Under a Cumulative Damage Process With Replacements

Zacks (Failure distribution associated with general renewal damage processes. In: Nikulin M, Commenges D, Haber C (eds) Probability statistics and modelling in public health. Springer, Berlin, pp 465–475, 2006) studied the reliability function, the hazard function and the distribution of the failure time when a system is subject to a cumulative, compound renewal damage process. The failure occurs when the damage process crosses a threshold β. In the present paper these results are generalized to the model where the system is replaced after failures. Two cases are considered: instant replacement and random positive replacement time. The distribution of the age of the current renewal cycle, as well as its excess life, and the availability function are studied. We derive also the distribution of total time in (0, t) at which the system has been operational.     

On a class of renewal risk model with random income

In this paper, we consider a renewal risk process with random premium income based on a Poisson process. Generating function for the discounted penalty function is obtained. We show that the discounted penalty function satisfies a defective renewal equation and the corresponding explicit expression can be obtained via a compound geometric tail. Finally, we consider the Laplace transform of the time to ruin, and derive the closed‐form expression for it when the claims have a discrete K_m distribution (i.e. the generating function of the distribution function is a ratio of two polynomials of order m∈?⁺). Copyright © 2008 John Wiley & Sons, Ltd.     

Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model

Consider a compound renewal risk model, in which a single accident may cause more than one claim. Under the condition that the common distribution of the individual claims is second order subexponential, we establish a second order asymptotic formula for the infinite-time ruin probability. Compared with the traditional ones, our second order asymptotic result is more precise and effective, which can be demonstrated by the numerical studies.

     

Zur Überlagerung von Erneuerungsprozessen

Summary If a finite sequence of independent (not necessarily stationary) renewal processes is given, a superposition process can easily be defined as the union of all point sequences represented by the given processes. The properties of such superposition processes are investigated. First, a necessary and sufficient condition for a superposition process to be a renewal process is given. Essentially, this condition reads thus: the given processes must be Poisson processes. The main result given in this paper is a limit theorem for superposition processes which shows that, even with largely arbitrary renewal processes superimposed, the superposition process has local properties which approach the properties of the Poisson process as the number of given processes increases. The theorem contains some well-known special theorems of this type [e.g. Khintchine, 1960; Franken, 1963].

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Von der Fakultät für Allgemeine Wissenschaften der T. H. München angenommene Habilitationsschrift (Auszug).     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

Busy period,virtual waiting time and number of customers in G<Superscript><Emphasis Type="Italic">δ</Emphasis></Superscript>|M<Superscript><Emphasis Type="Italic">ϰ</Emphasis></Superscript>|1|B system

In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected at its infimum. We use the results obtained to study a G ^δ |M ^ϰ |1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.     

连续时间复合二项模型多维精算量的联合分布

连续时间复合二项模型是由文献首先提出的.作为离散时间复合二项模型的连续化版本,连续时间复合二项模型的极限形式即为经典风险模型.为了得到该模型多维精算量的联合分布,该文引入了一列上穿零点,推导出该列上穿零点所构成的缺陷(defective)更新序列的更新质量函数.利用此更新质量函数及余额过程的强马氏性可以得到破产概率和包含破产时间,破产前余额,破产严重程度,破产前最大盈余,破产到恢复的最大赤字,整个过程的最大赤字等多维精算量的联合分布.由此联合分布得到其1-骨架链—离散时间复合二项模型的对应的联合分布,最后给出在1-骨架链中索赔额服从指数分布时这一特殊情况下相应多维精算量的联合分布的明确表达式.     

On the Number of Lost Customers in Stationary Loss Systems in the Light Traffic Case

For the stationary loss systems M/M/m/K and GI/M/m/K, we study two quantities: the number of lost customers during the time interval (0,t] (the first system only), and the number of lost customers among the first n customers to arrive (both systems). We derive explicit bounds for the total variation distances between the distributions of these quantities and compound Poisson–geometric distributions. The bounds are small in the light traffic case, i.e., when the loss of a customer is a rare event. To prove our results, we show that the studied quantities can be interpreted as accumulated rewards of stationary renewal reward processes, embedded into the queue length process or the process of queue lengths immediately before arrivals of new customers, and apply general results by Erhardsson on compound Poisson approximation for renewal reward processes.     

The Distribution of Discounted Compound PH–Renewal Processes

We continue and extend the work in Léveillé et al. (2010) Scand Actuar J 3:165–184 that gives analytical formulas for the moment generating function (mgf) of some discounted compound renewal processes. Here these mgf’s are used to derive and study the distribution of discounted compound Phase-type (PH) renewal sums. The approach consists in first deriving a differential equation, in the time variable, for the mgf of discounted compound sums when the inter–arrival times are PH–distributed. Then the corresponding distribution of the discounted compound PH sum is obtained by inversion of its mgf (or the equivalent Laplace transform). Analytical expressions for the asymptotic distribution of these discounted compound renewal sums are also given, to test general expressions in some limiting cases. Despite the technical difficulty, several new examples are provided where the inversion is possible, through symbolic computation in MAPLE. When the inversion is too complex, a truncated series solution method is proposed, which is not new but turns out to be very fast and accurate here, much more than other methods (such as rational approximations). The article concludes with some applications where the mgf and the distribution are used to calculate risk measures such as VaR, CTE, Esscher’s, Wang’s Proportional Hazard transforms and evaluate their evolution in time.     

Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs

We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

     

Exact transient analysis of a planar random motion with three directions

Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.     

Fisher information in window censored renewal process data and its applications

Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.     

复合Poisson-Geometric风险模型Gerber-Shiu折现惩罚函数

本文研究赔付为复合Poisson-Geometric过程的风险模型,首先得到了Gerber-Shiu折现惩罚期望函数所满足的更新方程,然后在此基础上推导出了破产概率和破产即刻前赢余分布等所满足的更新方程,再运用Laplace方法得出了破产概率的Pollazek-Khinchin公式,最后根据Pollazek-Khinchin公式,直接得出了当索赔分布服从指数分布的情形下破产概率的显示表达式.     

A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States

We consider the stationary distribution of the M/GI/1 type queue when background states are countable. We are interested in its tail behavior. To this end, we derive a Markov renewal equation for characterizing the stationary distribution using a Markov additive process that describes the number of customers in system when the system is not empty. Variants of this Markov renewal equation are also derived. It is shown that the transition kernels of these renewal equations can be expressed by the ladder height and the associated background state of a dual Markov additive process. Usually, matrix analysis is extensively used for studying the M/G/1 type queue. However, this may not be convenient when the background states are countable. We here rely on stochastic arguments, which not only make computations possible but also reveal new features. Those results are applied to study the tail decay rates of the stationary distributions. This includes refinements of the existence results with extensions.