Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair...Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.展开更多
Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {（Xi, Yi）, i≥1） form a seq...Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {（Xi, Yi）, i≥1） form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.展开更多
This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the different...This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.展开更多
We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same wa...We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same way. We derive the Laplace transform for the first passage time to surplus zero from a given negative surplus and for the duration of negative surplus. Closed-form expressions are given in the case of exponential individual claim. Finally, numerical results are provided to show how to estimate the moments of duration of negative surplus.展开更多
Based on hourly precipitation data in eastern China in the warm season during 1961-2000,spatial distributions of frequency for 20 mm h 1 and 50 mm h 1 precipitation were analyzed,and the criteria of short-duration rai...Based on hourly precipitation data in eastern China in the warm season during 1961-2000,spatial distributions of frequency for 20 mm h 1 and 50 mm h 1 precipitation were analyzed,and the criteria of short-duration rainfall events and severe rainfall events are discussed.Furthermore,the percentile method was used to define local hourly extreme precipitation;based on this,diurnal variations and trends in extreme precipitation were further studied.The results of this study show that,over Yunnan,South China,North China,and Northeast China,the most frequent extreme precipitation events occur most frequently in late afternoon and/or early evening.In the Guizhou Plateau and the Sichuan Basin,the maximum frequency of extreme precipitation events occurs in the late night and/or early morning.And in the western Sichuan Plateau,the maximum frequency occurs in the middle of the night.The frequency of extreme precipitation （based on hourly rainfall measurements） has increased in most parts of eastern China,especially in Northeast China and the middle and lower reaches of the Yangtze River,but precipitation has decreased significantly in North China in the past 50 years.In addition,stations in the Guizhou Plateau and the middle and lower reaches of the Yangtze River exhibit significant increasing trends in hourly precipitation extremes during the nighttime more than during the daytime.展开更多
Qinhuangdao City is located in the mid-latitude monsoon-affected region,and the timing of sea-ice coverage changes from year to year,making sea-ice forecasting difficult.In this paper,we propose a statistical model us...Qinhuangdao City is located in the mid-latitude monsoon-affected region,and the timing of sea-ice coverage changes from year to year,making sea-ice forecasting difficult.In this paper,we propose a statistical model using the 1980-2013 data collected at the Qinhuangdao observation station.The start date and the duration of ice coverage are fitted with four marginal distributions,from which the best-fitted,i.e.,the Weibull distribution,is selected to form a joint probability density function(PDF),built by the Gaussian copula method,for the two variables.With a given start date forecast by the Gray-Markov model(GMM),the joint PDF becomes a conditional probability model,which predicts that the duration of ice coverage is most likely 33 days at the Qinhuangdao observation station in 2014-2015.The predicted duration value is only two days less than the actual situation.The results prove that the new prediction model is feasible and effective to predict the period of ice coverage.The general sea-ice conditions that the sea ice would most likely form on December 8 and last for 80 days at the Qinhuangdao observation station could also be obtained from the joint PDF.The statistical model provides a useful tool to forecast ice conditions for planning and management of maritime activities.展开更多
In this paper, the ruin distributions were analyzed, including the distribution of surplus immediately before ruin, the distribution of claim at the time of ruin, the distribution of deficit, and the distribution of s...In this paper, the ruin distributions were analyzed, including the distribution of surplus immediately before ruin, the distribution of claim at the time of ruin, the distribution of deficit, and the distribution of surplus at the beginning of the claim period before ruin. Several integral equations for the ruin distributions were derived and some solutions under special conditions were obtained.展开更多
Project Evaluation and Review Technique (PERT) alongside recent modifications is a popular and useful tool in project risk analysis. Over the past seven decades, there have been some modifications in PERT owing to the...Project Evaluation and Review Technique (PERT) alongside recent modifications is a popular and useful tool in project risk analysis. Over the past seven decades, there have been some modifications in PERT owing to the shift from beta distributed activity times to other activity time distributions. This paper presents a review of activity time distributions in risk analysis as found in literature up to date.展开更多
Sunshine duration has an important impact on agriculture.In order to study the temporal and spatial variation of sunshine duration in China under the background of climate change,based on the observation data of 2089 ...Sunshine duration has an important impact on agriculture.In order to study the temporal and spatial variation of sunshine duration in China under the background of climate change,based on the observation data of 2089 national meteorological stations during 1961-2017,trend analysis,mutation analysis,partial correlation analysis,and Mann-Kendall mutation analysis were used to analyze the temporal and spatial variation characteristics of sunshine duration in different regions of China,and the influence of main climatic factors and human activities.The results showed that:the sunshine duration in China showed a significant decreasing trend with a rate of-45.8 h/10 a,and the sunshine duration in 7 regions of Northwestern China,Northern China,Northeastern China,Center China,Eastern China,Southern China,and Southwestern China also showed a significant decrease.The spatial distribution of sunshine duration in China was characterized by"less in the south and more in the north",and the sunshine duration in the northern region was significantly higher than those in the southern region.The sunshine duration in Tibet,Qinghai,Gansu and west Inner Mongolia was higher,while it was obviously lower in Sichuan Basin.Through M-K mutation test analysis,it was found that sunshine duration in the whole country decreased significantly,but there was no mutation station,while mutation occurred at 1989 in Southwestern China,1983 in Northwestern China,1985 in Northeastern China.Seasonally,there was the highest sunshine duration in summer,followed by spring,autumn,and winter;the decline rate was also the highest in summer,followed by autumn,winter,and spring.Sunshine duration had a highly negative correlation with total cloud amount,visibility,haze,and precipitation,while had a strongly positive correlation with wind speed,and had no significant correlation with fog.展开更多
This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process.Under the assumption that the claim-number process satisfies the large deviations principle,the uniform asymptoti...This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process.Under the assumption that the claim-number process satisfies the large deviations principle,the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes.Further,as an application of the result of onedimensional risk model,we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.展开更多
The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim ...The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim amount distribution is a finite mixture of exponential distributions or a Gamma (2, α) distribution.展开更多
Characterization of a mobile radio channel plays an important role in designing a reliable wireless communication system. Such channels are analyzed by two state model, namely satisfactory and outage state. This paper...Characterization of a mobile radio channel plays an important role in designing a reliable wireless communication system. Such channels are analyzed by two state model, namely satisfactory and outage state. This paper presents the analysis to estimate fading parameters of wireless channel with omission of certain outage durations which are considered as “Tolerance time”. Minimum outage duration which can be tolerated by a wireless fading channel to achieve desired packet error rate is defined as tolerance time. Normally a system with tolerable minimum outage time is analyzed based on Fade Duration Distribution (FDD) function over Rayleigh channel. In this paper Weibull function is used as FDD for varying tolerance time. The approach is simple and in general applicable from Rayleigh to Nakagami channels. The analysis is extended to study the effect of Tolerance time on channel fading statistics such as Average Fade Duration (AFD) and frequency of outage. Further the effects of various fade margin and Doppler spread on fading parameters are also investigated. The analysis can also be used in case of timeout expiration, connection resetting and congestion window control.展开更多
The compound negative binomial model, introduced in this paper, is a discrete time version. We discuss the Markov properties of the surplus process, and study the ruin probability and the joint distributions of actuar...The compound negative binomial model, introduced in this paper, is a discrete time version. We discuss the Markov properties of the surplus process, and study the ruin probability and the joint distributions of actuarial random vectors in this model. By the strong Markov property and the mass function of a defective renewal sequence, we obtain the explicit expressions of the ruin probability, the finite-horizon ruin probability, the joint distributions of T, U（T - 1）, ｜U（T）｜ and inf U（n） （i.e., the time of ruin, the surplus immediately before ruin, the deficit at ruin and maximal deficit from ruin to recovery） and the distributions of some actuarial random vectors.展开更多
Short-duration heavy rainfall（SDHR） is a type of severe convective weather that often leads to substantial losses of property and life. We derive the spatiotemporal distribution and diurnal variation of SDHR over Ch...Short-duration heavy rainfall（SDHR） is a type of severe convective weather that often leads to substantial losses of property and life. We derive the spatiotemporal distribution and diurnal variation of SDHR over China during the warm season（April–September） from quality-controlled hourly raingauge data taken at 876 stations for 19 yr（1991–2009）, in comparison with the diurnal features of the mesoscale convective systems（MCSs） derived from satellite data. The results are as follows. 1） Spatial distributions of the frequency of SDHR events with hourly rainfall greater than 10–40 mm are very similar to the distribution of heavy rainfall（daily rainfall 50 mm） over China's Mainland. 2） SDHR occurs most frequently in South China such as southern Yunnan, Guizhou, and Jiangxi provinces, the Sichuan basin, and the lower reaches of the Yangtze River, among others. Some SDHR events with hourly rainfall 50 mm also occur in northern China, e.g., the western Xinjiang and central-eastern Inner Mongolia. The heaviest hourly rainfall is observed over the Hainan Island with the amount reaching over 180 mm. 3） The frequency of the SDHR events is the highest in July, followed by August. Analysis of pentad variations in SDHR reveals that SDHR events are intermittent, with the fourth pentad of July the most active. The frequency of SDHR over China's Mainland increases slowly with the advent of the East Asian summer monsoon, but decreases rapidly with its withdrawal. 4） The diurnal peak of the SDHR activity occurs in the later afternoon（1600–1700 Beijing Time（BT））, and the secondary peak occurs after midnight（0100–0200 BT） and in the early morning（0700–0800 BT）; whereas the diurnal minimum occurs around late morning till noon（1000–1300 BT）. 5） The diurnal variation of SDHR exhibits generally consistent features with that of the MCSs in China, but the active periods and propagation of SDHR and MCSs difer in diferent regions. The number and duration of local maxima in the diurnal cycles of SDHR and MCSs also vary by region, with single, double, and even multiple peaks in some cases. These variations may be associated with the diferences in large-scale atmospheric circulation, surface conditions, and land-sea distribution.展开更多
In this paper we first consider a risk process in which claim inter-arrival times and the time until the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate rui...In this paper we first consider a risk process in which claim inter-arrival times and the time until the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate ruin, given an initial reserve of u when the claim size follows a Pareto distribution. Follow Ramsay[8], Laplace transforms and exponential integrals are used to derive the solution, which involves a single integral of real valued functions along the positive real line, and the integrand is not of an oscillating kind. Then we show that the ultimate ruin probability can be expressed as the sum of expected values of functions of two different Gamma random variables. Finally, the results are extended to the Erlang(n) case. Numerical examples are given to illustrate the main results.展开更多
基金Supported by the Natural Science Foundation of China(12071487,11671404)the Natural Science Foundation of Anhui Province(2208085MA06)+1 种基金the Provincial Natural Science Research Project of Anhui Colleges(KJ2021A0049,KJ2021A0060)Hunan Provincial Innovation Foundation for Postgraduate(CX20200146)。
文摘Consider a nonstandard continuous-time bidimensional risk model with constant force of interest,in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent.Under some mild conditions,we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval.If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed,it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval.
基金Supported by the National Natural Science Foundation of China(11001052,11171065,11326175)China Postdoctoral Science Foundation(2012M520964)+2 种基金Natural Science Foundation of Jiangsu Province ofChina(BK20131339)Postdoctoral Research Program of Jiangsu Province(1302015C)Qing Lan Project and Project of Construction for Superior Subjects of Statistics&Audit Science and Technology of Jiangsu Higher Education Institutions
文摘Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {（Xi, Yi）, i≥1） form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.
文摘This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.
基金Supported in part by the National Natural Science Foundation of China and the Ministry of Education of China
文摘We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same way. We derive the Laplace transform for the first passage time to surplus zero from a given negative surplus and for the duration of negative surplus. Closed-form expressions are given in the case of exponential individual claim. Finally, numerical results are provided to show how to estimate the moments of duration of negative surplus.
基金supported by the R & D Special Fund for Public Welfare Industry (meteorology)(GYHY201106018)National Key Program for Developing Basic Sciences (Grant No. 2006CB400503)
文摘Based on hourly precipitation data in eastern China in the warm season during 1961-2000,spatial distributions of frequency for 20 mm h 1 and 50 mm h 1 precipitation were analyzed,and the criteria of short-duration rainfall events and severe rainfall events are discussed.Furthermore,the percentile method was used to define local hourly extreme precipitation;based on this,diurnal variations and trends in extreme precipitation were further studied.The results of this study show that,over Yunnan,South China,North China,and Northeast China,the most frequent extreme precipitation events occur most frequently in late afternoon and/or early evening.In the Guizhou Plateau and the Sichuan Basin,the maximum frequency of extreme precipitation events occurs in the late night and/or early morning.And in the western Sichuan Plateau,the maximum frequency occurs in the middle of the night.The frequency of extreme precipitation （based on hourly rainfall measurements） has increased in most parts of eastern China,especially in Northeast China and the middle and lower reaches of the Yangtze River,but precipitation has decreased significantly in North China in the past 50 years.In addition,stations in the Guizhou Plateau and the middle and lower reaches of the Yangtze River exhibit significant increasing trends in hourly precipitation extremes during the nighttime more than during the daytime.
基金support of the National Natural Science Foundation of China (No. 51779236)the NSFC-Shandong Joint Fund Project (No. U1706226)the Ocean University of China under the support of 111 Project (No. B14028)
文摘Qinhuangdao City is located in the mid-latitude monsoon-affected region,and the timing of sea-ice coverage changes from year to year,making sea-ice forecasting difficult.In this paper,we propose a statistical model using the 1980-2013 data collected at the Qinhuangdao observation station.The start date and the duration of ice coverage are fitted with four marginal distributions,from which the best-fitted,i.e.,the Weibull distribution,is selected to form a joint probability density function(PDF),built by the Gaussian copula method,for the two variables.With a given start date forecast by the Gray-Markov model(GMM),the joint PDF becomes a conditional probability model,which predicts that the duration of ice coverage is most likely 33 days at the Qinhuangdao observation station in 2014-2015.The predicted duration value is only two days less than the actual situation.The results prove that the new prediction model is feasible and effective to predict the period of ice coverage.The general sea-ice conditions that the sea ice would most likely form on December 8 and last for 80 days at the Qinhuangdao observation station could also be obtained from the joint PDF.The statistical model provides a useful tool to forecast ice conditions for planning and management of maritime activities.
文摘In this paper, the ruin distributions were analyzed, including the distribution of surplus immediately before ruin, the distribution of claim at the time of ruin, the distribution of deficit, and the distribution of surplus at the beginning of the claim period before ruin. Several integral equations for the ruin distributions were derived and some solutions under special conditions were obtained.
文摘Project Evaluation and Review Technique (PERT) alongside recent modifications is a popular and useful tool in project risk analysis. Over the past seven decades, there have been some modifications in PERT owing to the shift from beta distributed activity times to other activity time distributions. This paper presents a review of activity time distributions in risk analysis as found in literature up to date.
基金Supported by the National Key R&D Plan(2017YFD0300201,2020YFE 0201900).
文摘Sunshine duration has an important impact on agriculture.In order to study the temporal and spatial variation of sunshine duration in China under the background of climate change,based on the observation data of 2089 national meteorological stations during 1961-2017,trend analysis,mutation analysis,partial correlation analysis,and Mann-Kendall mutation analysis were used to analyze the temporal and spatial variation characteristics of sunshine duration in different regions of China,and the influence of main climatic factors and human activities.The results showed that:the sunshine duration in China showed a significant decreasing trend with a rate of-45.8 h/10 a,and the sunshine duration in 7 regions of Northwestern China,Northern China,Northeastern China,Center China,Eastern China,Southern China,and Southwestern China also showed a significant decrease.The spatial distribution of sunshine duration in China was characterized by"less in the south and more in the north",and the sunshine duration in the northern region was significantly higher than those in the southern region.The sunshine duration in Tibet,Qinghai,Gansu and west Inner Mongolia was higher,while it was obviously lower in Sichuan Basin.Through M-K mutation test analysis,it was found that sunshine duration in the whole country decreased significantly,but there was no mutation station,while mutation occurred at 1989 in Southwestern China,1983 in Northwestern China,1985 in Northeastern China.Seasonally,there was the highest sunshine duration in summer,followed by spring,autumn,and winter;the decline rate was also the highest in summer,followed by autumn,winter,and spring.Sunshine duration had a highly negative correlation with total cloud amount,visibility,haze,and precipitation,while had a strongly positive correlation with wind speed,and had no significant correlation with fog.
基金Supported by the 333 High Level Talent Training Project of Jiangsu Provincethe National Natural Science Foundation of China(71871046)Science and Technology Projects of Sichuan Province(2021YFQ0007)。
文摘This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process.Under the assumption that the claim-number process satisfies the large deviations principle,the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes.Further,as an application of the result of onedimensional risk model,we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.
文摘The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim amount distribution is a finite mixture of exponential distributions or a Gamma (2, α) distribution.
文摘Characterization of a mobile radio channel plays an important role in designing a reliable wireless communication system. Such channels are analyzed by two state model, namely satisfactory and outage state. This paper presents the analysis to estimate fading parameters of wireless channel with omission of certain outage durations which are considered as “Tolerance time”. Minimum outage duration which can be tolerated by a wireless fading channel to achieve desired packet error rate is defined as tolerance time. Normally a system with tolerable minimum outage time is analyzed based on Fade Duration Distribution (FDD) function over Rayleigh channel. In this paper Weibull function is used as FDD for varying tolerance time. The approach is simple and in general applicable from Rayleigh to Nakagami channels. The analysis is extended to study the effect of Tolerance time on channel fading statistics such as Average Fade Duration (AFD) and frequency of outage. Further the effects of various fade margin and Doppler spread on fading parameters are also investigated. The analysis can also be used in case of timeout expiration, connection resetting and congestion window control.
基金Supported by the National Natural Science Foundation of China (No.10671197)
文摘The compound negative binomial model, introduced in this paper, is a discrete time version. We discuss the Markov properties of the surplus process, and study the ruin probability and the joint distributions of actuarial random vectors in this model. By the strong Markov property and the mass function of a defective renewal sequence, we obtain the explicit expressions of the ruin probability, the finite-horizon ruin probability, the joint distributions of T, U（T - 1）, ｜U（T）｜ and inf U（n） （i.e., the time of ruin, the surplus immediately before ruin, the deficit at ruin and maximal deficit from ruin to recovery） and the distributions of some actuarial random vectors.
基金Supported by the China Meteorological Administration Special Public Welfare Research Fund (GYHY201206004,GYHY201206003,and GYHY200906003)National (Key) Basic Research and Development (973) Program of China (2013CB430106)
文摘Short-duration heavy rainfall（SDHR） is a type of severe convective weather that often leads to substantial losses of property and life. We derive the spatiotemporal distribution and diurnal variation of SDHR over China during the warm season（April–September） from quality-controlled hourly raingauge data taken at 876 stations for 19 yr（1991–2009）, in comparison with the diurnal features of the mesoscale convective systems（MCSs） derived from satellite data. The results are as follows. 1） Spatial distributions of the frequency of SDHR events with hourly rainfall greater than 10–40 mm are very similar to the distribution of heavy rainfall（daily rainfall 50 mm） over China's Mainland. 2） SDHR occurs most frequently in South China such as southern Yunnan, Guizhou, and Jiangxi provinces, the Sichuan basin, and the lower reaches of the Yangtze River, among others. Some SDHR events with hourly rainfall 50 mm also occur in northern China, e.g., the western Xinjiang and central-eastern Inner Mongolia. The heaviest hourly rainfall is observed over the Hainan Island with the amount reaching over 180 mm. 3） The frequency of the SDHR events is the highest in July, followed by August. Analysis of pentad variations in SDHR reveals that SDHR events are intermittent, with the fourth pentad of July the most active. The frequency of SDHR over China's Mainland increases slowly with the advent of the East Asian summer monsoon, but decreases rapidly with its withdrawal. 4） The diurnal peak of the SDHR activity occurs in the later afternoon（1600–1700 Beijing Time（BT））, and the secondary peak occurs after midnight（0100–0200 BT） and in the early morning（0700–0800 BT）; whereas the diurnal minimum occurs around late morning till noon（1000–1300 BT）. 5） The diurnal variation of SDHR exhibits generally consistent features with that of the MCSs in China, but the active periods and propagation of SDHR and MCSs difer in diferent regions. The number and duration of local maxima in the diurnal cycles of SDHR and MCSs also vary by region, with single, double, and even multiple peaks in some cases. These variations may be associated with the diferences in large-scale atmospheric circulation, surface conditions, and land-sea distribution.
基金Supported by Postdoctoral Scientific Foundation of China,a CRGC grant from the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region,China (Project No.HKU 7139/01H).
文摘In this paper we first consider a risk process in which claim inter-arrival times and the time until the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate ruin, given an initial reserve of u when the claim size follows a Pareto distribution. Follow Ramsay[8], Laplace transforms and exponential integrals are used to derive the solution, which involves a single integral of real valued functions along the positive real line, and the integrand is not of an oscillating kind. Then we show that the ultimate ruin probability can be expressed as the sum of expected values of functions of two different Gamma random variables. Finally, the results are extended to the Erlang(n) case. Numerical examples are given to illustrate the main results.
