Pub Date : 2022-05-20DOI: 10.1017/s0269964822000134
Soumaya Idriss
In this paper, we treat a nonlinear and unbalanced $2$ -color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.
{"title":"Nonlinear and unbalanced urn models with two types of strategies: a stochastic approximation point of view","authors":"Soumaya Idriss","doi":"10.1017/s0269964822000134","DOIUrl":"https://doi.org/10.1017/s0269964822000134","url":null,"abstract":"\u0000 In this paper, we treat a nonlinear and unbalanced \u0000 \u0000 \u0000 $2$\u0000 \u0000 -color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"105 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82088535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-02DOI: 10.1017/s0269964822000523
K. R. Bhutani, Ravi Kalpathy, H. Mahmoud
We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.
{"title":"Random multi-hooking networks","authors":"K. R. Bhutani, Ravi Kalpathy, H. Mahmoud","doi":"10.1017/s0269964822000523","DOIUrl":"https://doi.org/10.1017/s0269964822000523","url":null,"abstract":"\u0000 We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"15 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88698274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1017/s0269964822000110
{"title":"PES volume 36 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0269964822000110","DOIUrl":"https://doi.org/10.1017/s0269964822000110","url":null,"abstract":"","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"24 1","pages":"b1 - b2"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78206532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1017/s0269964822000109
{"title":"PES volume 36 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0269964822000109","DOIUrl":"https://doi.org/10.1017/s0269964822000109","url":null,"abstract":"","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"67 1","pages":"f1 - f2"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74886606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-31DOI: 10.1017/S0269964822000080
Eric C. K. Cheung, Haibo Liu
This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.
本文研究了Gerber-Shiu期望折现惩罚函数[Gerber and Shiu(1998)]的推广。论毁灭的时间价值。北美精算杂志2(1):48-72]在扰动复合泊松保险风险模型的背景下，其中也包括了总贴现索赔和贴现小波动(由布朗运动引起)的矩，直到破产。特别是，后一个量由随机积分表示，据我们所知，在文献中从未对其进行过分析。导出了由广义Gerber-Shiu函数满足的递推积分-微分方程，并将其转化为有缺陷的更新方程。当单个索赔金额作为指数组合分布时，给出显式解决方案。给出了数值实例，包括贴现索赔与贴现扰动之间的协方差计算，直至破产。
{"title":"Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion","authors":"Eric C. K. Cheung, Haibo Liu","doi":"10.1017/S0269964822000080","DOIUrl":"https://doi.org/10.1017/S0269964822000080","url":null,"abstract":"This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"17 1","pages":"387 - 417"},"PeriodicalIF":1.1,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87250436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-15DOI: 10.1017/S0269964821000504
G. S. Fishman, S. Stidham
A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.
{"title":"An adaptive strategy for offering m-out-of-n insurance policies","authors":"G. S. Fishman, S. Stidham","doi":"10.1017/S0269964821000504","DOIUrl":"https://doi.org/10.1017/S0269964821000504","url":null,"abstract":"A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"1 1","pages":"106 - 134"},"PeriodicalIF":1.1,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86470075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1017/s0269964822000043
M'hamed Gaïgi, Idris Kharroubi, T. Lim
In this paper, we consider the problem of sustainable harvesting. We explain how the manager maximizes his/her profit according to the quantity of natural resource available in a harvesting area and under the constraint of penalties and fines when the quota is exceeded. We characterize the optimal values and some optimal strategies using a verification result. We then show by numerical examples that this optimal strategy is better than naive ones. Moreover, we define a level of fines which insures the double objective of the sustainable harvesting: a remaining quantity of available natural resource to insure its sustainability and an acceptable income for the manager.
{"title":"Optimal management and valuation of a natural resource: the case of optimal harvesting","authors":"M'hamed Gaïgi, Idris Kharroubi, T. Lim","doi":"10.1017/s0269964822000043","DOIUrl":"https://doi.org/10.1017/s0269964822000043","url":null,"abstract":"\u0000 In this paper, we consider the problem of sustainable harvesting. We explain how the manager maximizes his/her profit according to the quantity of natural resource available in a harvesting area and under the constraint of penalties and fines when the quota is exceeded. We characterize the optimal values and some optimal strategies using a verification result. We then show by numerical examples that this optimal strategy is better than naive ones. Moreover, we define a level of fines which insures the double objective of the sustainable harvesting: a remaining quantity of available natural resource to insure its sustainability and an acceptable income for the manager.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86695968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-22DOI: 10.1017/S0269964822000018
Yu Chen, Yu Xing
Abstract In this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.
{"title":"Credit default swap pricing with counterparty risk in a reduced form model with a common jump process","authors":"Yu Chen, Yu Xing","doi":"10.1017/S0269964822000018","DOIUrl":"https://doi.org/10.1017/S0269964822000018","url":null,"abstract":"Abstract In this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"56 1","pages":"275 - 293"},"PeriodicalIF":1.1,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91298464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-21DOI: 10.1017/S0269964822000031
K. Zografos
This paper concentrates on the fundamental concepts of entropy, information and divergence to the case where the distribution function and the respective survival function play the central role in their definition. The main aim is to provide an overview of these three categories of measures of information and their cumulative and survival counterparts. It also aims to introduce and discuss Csiszár's type cumulative and survival divergences and the analogous Fisher's type information on the basis of cumulative and survival functions.
{"title":"On reconsidering entropies and divergences and their cumulative counterparts: Csiszár's, DPD's and Fisher's type cumulative and survival measures","authors":"K. Zografos","doi":"10.1017/S0269964822000031","DOIUrl":"https://doi.org/10.1017/S0269964822000031","url":null,"abstract":"This paper concentrates on the fundamental concepts of entropy, information and divergence to the case where the distribution function and the respective survival function play the central role in their definition. The main aim is to provide an overview of these three categories of measures of information and their cumulative and survival counterparts. It also aims to introduce and discuss Csiszár's type cumulative and survival divergences and the analogous Fisher's type information on the basis of cumulative and survival functions.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"14 1","pages":"294 - 321"},"PeriodicalIF":1.1,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73511879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-16DOI: 10.1017/S0269964820000595
Sihan Ding, G. Koole
Abstract In this paper, we consider a two-stage call center staffing model. In the first stage, the interval staffing levels are set under arrival rate uncertainty. In the second stage, these initial staffing levels are corrected to the right value based on more precise arrival rate information. We show that this problem is of newsvendor type, where the costs are the initial staffing costs plus the second stage adaptation costs. We show that we should initially staff according to a quantile of the distributional forecast, rather than the mean. It is also shown that the errors in staffing are approximately linear in the forecasting errors. This leads to the conclusion that the weighted sum of errors should be the error measurement in call center forecasting, since minimizing, it minimizes the total staffing costs. In special cases where the costs are symmetric for over- and understaffing, this is equivalent to minimizing the weighted absolute percentage error.
{"title":"Optimal call center forecasting and staffing","authors":"Sihan Ding, G. Koole","doi":"10.1017/S0269964820000595","DOIUrl":"https://doi.org/10.1017/S0269964820000595","url":null,"abstract":"Abstract In this paper, we consider a two-stage call center staffing model. In the first stage, the interval staffing levels are set under arrival rate uncertainty. In the second stage, these initial staffing levels are corrected to the right value based on more precise arrival rate information. We show that this problem is of newsvendor type, where the costs are the initial staffing costs plus the second stage adaptation costs. We show that we should initially staff according to a quantile of the distributional forecast, rather than the mean. It is also shown that the errors in staffing are approximately linear in the forecasting errors. This leads to the conclusion that the weighted sum of errors should be the error measurement in call center forecasting, since minimizing, it minimizes the total staffing costs. In special cases where the costs are symmetric for over- and understaffing, this is equivalent to minimizing the weighted absolute percentage error.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"26 1","pages":"254 - 263"},"PeriodicalIF":1.1,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85288626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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