NEW RESULTS ON THE DISTRIBUTION OF DISCOUNTED COMPOUND POISSON SUMS

2018 ◽  
Vol 49 (1) ◽  
pp. 169-187 ◽  
Author(s):  
Zhehao Zhang
Keyword(s):  
Infinite Time ◽  
Catastrophe Insurance ◽  
Compound Poisson ◽  
Gamma Distributions ◽  
Time Period ◽  
Heavy Tailed

AbstractThis paper focuses on the distribution of Poisson sums of discounted claims over a finite or infinite time period. It gives two new results when claim amounts follow Mittag-Leffler distributions and two new results when claim amounts follow gamma distributions. Further, as Mittag-Leffler distribution is of heavy-tailed nature and its moments only exist for order strictly smaller than one, this distribution can be used for modelling insurance whose claim amounts are extremely large, that is, catastrophe insurance.

Impact of the time pattern of human behaviors on information spreading

2014 ◽  
Vol 25 (11) ◽  
pp. 1450063 ◽  
Author(s):  
Lin Zhang ◽  
Tingting Pan ◽  
Ye Wu ◽  
Jinghua Xiao
Keyword(s):  
Activity Patterns ◽  
Sufficient Evidence ◽  
Time Pattern ◽  
Human Dynamics ◽  
Human Behaviors ◽  
Event Time ◽  
Time Period ◽  
Collective Levels ◽  
Information Spreading ◽  
Heavy Tailed

There is sufficient evidence to support that the inter-event time distribution of human behaviors is heavy tailed. This fact plays an important role in human dynamics at collective levels. Recently, researchers found that the heavy-tailed inter-activity patterns of human behaviors slow down the spreading significantly. In this paper, we investigate the influence of people's interest time length on information spreading. Considering that information has its own time period of being interested in, we find that for small interest time period, heavy-tailed human behaviors can enhance the information spreading. While for large interest time period, the heavy-tailed human behaviors will suppress the spreading. These results can help us understanding the rule of information spreading, controlling and limiting the outbreak of rumors.

scholarly journals Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments

2004 ◽  
Vol 36 (4) ◽  
pp. 1278-1299 ◽  
Author(s):  
Qihe Tang ◽  
Gurami Tsitsiashvili
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Economic Environment ◽  
Ruin Probabilities ◽  
Insurance Risk ◽  
Infinite Time ◽  
Time Period ◽  
Stochastic Returns ◽  
Regularly Varying

This paper investigates the finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly-varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, and so apply in the case of infinite-time ruin.

The Ruin Probability During an Infinite Time Period

Risk Theory ◽  
1977 ◽  
pp. 137-159
Author(s):  
Robert Eric Beard ◽  
Teivo Pentikäinen ◽  
Erkki Pesonen
Keyword(s):  
Ruin Probability ◽  
Infinite Time ◽  
Time Period

scholarly journals Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

2012 ◽  
Vol 49 (4) ◽  
pp. 939-953 ◽  
Author(s):  
Xuemiao Hao ◽  
Qihe Tang
Keyword(s):  
Risk Model ◽  
Moment Condition ◽  
Ruin Probabilities ◽  
Economic Factors ◽  
Infinite Time ◽  
Lévy Measure ◽  
Loss Process ◽  
Surplus Process ◽  
Heavy Tailed ◽  
Extended Regular Variation

Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y0=x > 0, evolves according to dYt= Yt- dRt -dPt for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.

scholarly journals Optimal Dynamic Reinsurance

2006 ◽  
Vol 36 (2) ◽  
pp. 415-432
Author(s):  
David C.M. Dickson ◽  
Howard R. Waters
Keyword(s):  
Poisson Process ◽  
Time Horizon ◽  
Optimal Strategies ◽  
Compound Poisson Process ◽  
Probability Of Ruin ◽  
Compound Poisson ◽  
Surplus Process ◽  
Gamma Distributions ◽  
Optimal Dynamic ◽  
Aggregate Claims

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

Evaluating the clustering of Central European rain-on-snow events with flood-inducing potential

2021 ◽  
Author(s):  
Moritz Johannes Kirschner ◽  
Amelie Krug ◽  
Lun David ◽  
Bodo Ahrens
Keyword(s):  
Stationary Process ◽  
Scan Statistics ◽  
Central European ◽  
Gamma Distributions ◽  
Time Period ◽  
Random Events ◽  
The Face ◽  
Major River ◽  
Clustering Pattern ◽  
Snowmelt Time

<p>Rain-on-snow (ROS) floods are responsible for the overwhelming majority of floods affecting multiple major river basins simultaneously in Europe during the last century. These widespread floods have serious negative economical, social and ecological effects, and knowledge about their rate of occurrence is critical for future projections in the face of climate change.</p><p>Recent studies have shown that ROS events (with flood-inducing potential) in Europe increase and decrease based on the elevation range considered since 1950 and there appears to be a clustering pattern of flood-poor and flood-rich periods since 1900. Our goal is to analyze if these changes in frequency can be realistically described by a stationary process (or a combination thereof) or if there must be hidden time-dependent driving factors to explain the observed clustering. To test this theory we analyze a simulation for the time period 1901-2010 based on ERA-20C dynamically downscaled using a coupled RCM. We apply a method from scan statistics and confirm the existence of significant periods poor and rich in ROS events with regards to the reference condition of independent and identically distributed random events and present their position in time. The same procedure is applied to the ROS event constituents (rainfall and snowmelt), where we identify such periods in the rainfall, but not in the snowmelt time series. We construct a stochastic ROS model by modelling precipitation and snowmelt via stationary gamma distributions fitted to our data but are unable to reproduce the observed clustering behaviour using the combined signal.</p><p>This study confirms that the observed ROS floods in Central Europe are unlikely to be the result of stationary processes which hints at climate drivers for the compound rain-on-snow process in Europe.</p>

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

2012 ◽  
Vol 49 (04) ◽  
pp. 939-953
Author(s):  
Xuemiao Hao ◽  
Qihe Tang
Keyword(s):  
Risk Model ◽  
Moment Condition ◽  
Ruin Probabilities ◽  
Economic Factors ◽  
Infinite Time ◽  
Lévy Measure ◽  
Loss Process ◽  
Surplus Process ◽  
Heavy Tailed ◽  
Extended Regular Variation

Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y 0=x > 0, evolves according to dY t = Y t- dR t -dP t for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.

Optimal admission control for a single-server loss queue

2004 ◽  
Vol 41 (02) ◽  
pp. 535-546 ◽  
Author(s):  
Kyle Y. Lin ◽  
Sheldon M. Ross
Keyword(s):  
Admission Control ◽  
Time Horizon ◽  
Queueing System ◽  
Single Server ◽  
Sufficient Condition ◽  
Infinite Time ◽  
Discounted Cost ◽  
Time Period ◽  
Service Distribution ◽  
General Service

This paper presents a single-server loss queueing system where customers arrive according to a Poisson process. Upon arrival, the customer presents itself to a gatekeeper who has to decide whether to admit the customer into the system without knowing the busy–idle status of the server. There is a cost if the gatekeeper blocks a customer, and a larger cost if an admitted customer finds the server busy and therefore has to leave the system. The goal of the gatekeeper is to minimize the total expected discounted cost on an infinite time horizon. In the case of an exponential service distribution, we show that a threshold-type policy—block for a time period following each admission and then admit the next customer—is optimal. For general service distributions, we show that a threshold-type policy need not be optimal; we then present a sufficient condition for the existence of an optimal threshold-type policy.

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