Pub Date : 2022-01-11DOI: 10.1080/15326349.2021.2003712
A. Roitershtein, Zirou Zhou
Abstract We consider a history-dependent random linear recursion, adapting a continuous-time framework introduced by Clifford and Stirzaker. Typically, the main object of interest in the study of history-dependent processes is the evolution and asymptotic behavior of their first and second moments. We apply the methodology developed by Clifford and Stirzaker to study the evolution of distribution tails of the process by utilizing a certain affinity between the asymptotic structure of the tails in a regular variation regime and a linear structure of moments in the class of models introduced by Clifford and Stirzaker.
{"title":"Distribution tails of a history-dependent random linear recursion","authors":"A. Roitershtein, Zirou Zhou","doi":"10.1080/15326349.2021.2003712","DOIUrl":"https://doi.org/10.1080/15326349.2021.2003712","url":null,"abstract":"Abstract We consider a history-dependent random linear recursion, adapting a continuous-time framework introduced by Clifford and Stirzaker. Typically, the main object of interest in the study of history-dependent processes is the evolution and asymptotic behavior of their first and second moments. We apply the methodology developed by Clifford and Stirzaker to study the evolution of distribution tails of the process by utilizing a certain affinity between the asymptotic structure of the tails in a regular variation regime and a linear structure of moments in the class of models introduced by Clifford and Stirzaker.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47095139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-08DOI: 10.1080/15326349.2021.2011748
G. Bet, Jori Selen, Alessandro Zocca
Abstract We consider a model for a queue in which only a fixed number N of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov chain on a finite triangular region of the square lattice. We interpret the resulting random walk on as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather general otherwise. We untangle the resulting intricate combinatorial structure by introducing appropriate generating functions that exploit the recursive structure of the model. This allows us to derive an explicit expression for the probability mass function of the number of customers served in any busy period (equivalently, of the length of any excursion of the Dyck path above the diagonal) as a weighted sum with alternating sign over a certain subclass of Dyck paths.
{"title":"Weighted Dyck paths and nonstationary queues","authors":"G. Bet, Jori Selen, Alessandro Zocca","doi":"10.1080/15326349.2021.2011748","DOIUrl":"https://doi.org/10.1080/15326349.2021.2011748","url":null,"abstract":"Abstract We consider a model for a queue in which only a fixed number N of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov chain on a finite triangular region of the square lattice. We interpret the resulting random walk on as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather general otherwise. We untangle the resulting intricate combinatorial structure by introducing appropriate generating functions that exploit the recursive structure of the model. This allows us to derive an explicit expression for the probability mass function of the number of customers served in any busy period (equivalently, of the length of any excursion of the Dyck path above the diagonal) as a weighted sum with alternating sign over a certain subclass of Dyck paths.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44982917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-04DOI: 10.1080/15326349.2021.2018336
J. Cha, N. Limnios
{"title":"Discrete Time Minimal Repair Process and Its Reliability Applications under Random Environments","authors":"J. Cha, N. Limnios","doi":"10.1080/15326349.2021.2018336","DOIUrl":"https://doi.org/10.1080/15326349.2021.2018336","url":null,"abstract":"","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48648825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1080/15326349.2021.2015386
C. Smadi, V. Vatutin
Abstract We consider a two-type decomposable branching process where type 1 particles may produce particles of types 1 and 2 while type 2 particles can give birth only to type 2 particles. Let i = 1, 2 be the number of type i particles existing in the process at moment m < n and having a positive number of descendants at moment n. Assuming that particles of the first type evolve in a random environment and particles of the second type evolve in a constant environment we investigate the distribution of the random vector when and
{"title":"Reduced processes evolving in a mixed environment","authors":"C. Smadi, V. Vatutin","doi":"10.1080/15326349.2021.2015386","DOIUrl":"https://doi.org/10.1080/15326349.2021.2015386","url":null,"abstract":"Abstract We consider a two-type decomposable branching process where type 1 particles may produce particles of types 1 and 2 while type 2 particles can give birth only to type 2 particles. Let i = 1, 2 be the number of type i particles existing in the process at moment m < n and having a positive number of descendants at moment n. Assuming that particles of the first type evolve in a random environment and particles of the second type evolve in a constant environment we investigate the distribution of the random vector when and","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47314234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-21DOI: 10.1080/15326349.2021.2005628
E. T. Kolkovska, José Alfredo López Mimbela, José Hermenegildo Ramírez González
Abstract Let be a generating function, where are nonnegative measurable functions, and let be a continuous function. We prove that reaction-diffusion equations of the prototype possess nontrivial positive global solutions under suitable assumptions on and
{"title":"Existence of global solutions of a nonautonomous semilinear equation with varying reaction","authors":"E. T. Kolkovska, José Alfredo López Mimbela, José Hermenegildo Ramírez González","doi":"10.1080/15326349.2021.2005628","DOIUrl":"https://doi.org/10.1080/15326349.2021.2005628","url":null,"abstract":"Abstract Let be a generating function, where are nonnegative measurable functions, and let be a continuous function. We prove that reaction-diffusion equations of the prototype possess nontrivial positive global solutions under suitable assumptions on and","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46355806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-19DOI: 10.1080/15326349.2021.2006066
N. Brites, C. Braumann
Abstract We can describe the size evolution of a harvested population in a randomly varying environment using stochastic differential equations. Previously, we have compared the profit performance of four harvesting policies: (i) optimal variable effort policy, based on variable effort; (ii) optimal penalized variable effort policies, penalized versions based on including an artificial running energy cost on the effort; (iii) stepwise policies, staircase versions where the harvesting effort is determined at the beginning of each year (or of each biennium) and kept constant throughout that year (or biennium); (iv) constant harvesting effort sustainable policy, based on constant effort. They have different properties, so it is also worth looking at combinations of such policies and studying the single and cross-effects of the amount of penalization, the absence or presence and type of steps, and the restraints on minimum and maximum allowed efforts. Using data based on a real harvested population and considering a logistic growth model, we perform such a comparison study of pure and mixed policies in terms of profit, applicability, and other relevant properties. We end up answering the question: is the optimal enemy of the good?
{"title":"Harvesting optimization with stochastic differential equations models: is the optimal enemy of the good?","authors":"N. Brites, C. Braumann","doi":"10.1080/15326349.2021.2006066","DOIUrl":"https://doi.org/10.1080/15326349.2021.2006066","url":null,"abstract":"Abstract We can describe the size evolution of a harvested population in a randomly varying environment using stochastic differential equations. Previously, we have compared the profit performance of four harvesting policies: (i) optimal variable effort policy, based on variable effort; (ii) optimal penalized variable effort policies, penalized versions based on including an artificial running energy cost on the effort; (iii) stepwise policies, staircase versions where the harvesting effort is determined at the beginning of each year (or of each biennium) and kept constant throughout that year (or biennium); (iv) constant harvesting effort sustainable policy, based on constant effort. They have different properties, so it is also worth looking at combinations of such policies and studying the single and cross-effects of the amount of penalization, the absence or presence and type of steps, and the restraints on minimum and maximum allowed efforts. Using data based on a real harvested population and considering a logistic growth model, we perform such a comparison study of pure and mixed policies in terms of profit, applicability, and other relevant properties. We end up answering the question: is the optimal enemy of the good?","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48016426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-18DOI: 10.1080/15326349.2022.2066131
Miguel González, Pedro Martín-Chávez, I. D. del Puerto
Abstract A controlled branching process (CBP) is a modification of the standard Bienaymé–Galton–Watson process in which the number of progenitors in each generation is determined by a random mechanism. We consider a CBP starting from a random number of initial individuals. The main aim of this article is to provide a Feller diffusion approximation for critical CBPs. A similar result by considering a fixed number of initial individuals by using operator semigroup convergence theorems has been previously proved by Sriram et al. (Stochastic Processes Appl. 2007;117:928–946). An alternative proof is now provided making use of limit theorems for random step processes.
{"title":"Diffusion approximation of controlled branching processes using limit theorems for random step processes","authors":"Miguel González, Pedro Martín-Chávez, I. D. del Puerto","doi":"10.1080/15326349.2022.2066131","DOIUrl":"https://doi.org/10.1080/15326349.2022.2066131","url":null,"abstract":"Abstract A controlled branching process (CBP) is a modification of the standard Bienaymé–Galton–Watson process in which the number of progenitors in each generation is determined by a random mechanism. We consider a CBP starting from a random number of initial individuals. The main aim of this article is to provide a Feller diffusion approximation for critical CBPs. A similar result by considering a fixed number of initial individuals by using operator semigroup convergence theorems has been previously proved by Sriram et al. (Stochastic Processes Appl. 2007;117:928–946). An alternative proof is now provided making use of limit theorems for random step processes.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47242784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-02DOI: 10.1080/15326349.2022.2043166
S. Hautphenne, Minyuan Li
Abstract We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their deterministic (fluid) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approximation does not allow us to determine the time until extinction directly, we propose several methods to complement this approach. We also use the deterministic approach to study the behavior of the processes as we increase the magnitude of the individual’s birth rate.
{"title":"A fluid approach to total-progeny-dependent birth-and-death processes","authors":"S. Hautphenne, Minyuan Li","doi":"10.1080/15326349.2022.2043166","DOIUrl":"https://doi.org/10.1080/15326349.2022.2043166","url":null,"abstract":"Abstract We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their deterministic (fluid) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approximation does not allow us to determine the time until extinction directly, we propose several methods to complement this approach. We also use the deterministic approach to study the behavior of the processes as we increase the magnitude of the individual’s birth rate.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44712193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-20DOI: 10.1080/15326349.2021.1987264
Arindam Panja, Pradip Kundu, B. Pradhan
Abstract In this paper, we establish some stochastic comparison results for two finite mixture models where the corresponding random variables follow one of the parental families of distributions, namely, proportional odds, proportional hazards, and proportional reversed hazards. The results of this paper are illustrated with numerical examples.
{"title":"On stochastic comparisons of finite mixture models","authors":"Arindam Panja, Pradip Kundu, B. Pradhan","doi":"10.1080/15326349.2021.1987264","DOIUrl":"https://doi.org/10.1080/15326349.2021.1987264","url":null,"abstract":"Abstract In this paper, we establish some stochastic comparison results for two finite mixture models where the corresponding random variables follow one of the parental families of distributions, namely, proportional odds, proportional hazards, and proportional reversed hazards. The results of this paper are illustrated with numerical examples.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48578230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-17DOI: 10.1080/15326349.2021.1985520
Xiaowei Chen, F. Huang, Xiufang Li
Abstract This article describes a robust continuous-time asset-liability management problem under Markov regime-switching. First, we employ the “homothetic robustness” to preserve the performance of robustness for the ALM model, which runs well in precisely modified state variables and performs reasonably if some forms of model misspecification exist. Second, we consider the asset-to-liability ratio instead of the surplus, which ensures that we use relative values instead of absolute values to modify the wealth process. Besides, we use the stochastic dynamic programming method to get some closed-form results and analyze the impacts of parameters on the investment strategy and value function, respectively, by numerical examples.
{"title":"Robust asset-liability management under CRRA utility criterion with regime switching: a continuous-time model","authors":"Xiaowei Chen, F. Huang, Xiufang Li","doi":"10.1080/15326349.2021.1985520","DOIUrl":"https://doi.org/10.1080/15326349.2021.1985520","url":null,"abstract":"Abstract This article describes a robust continuous-time asset-liability management problem under Markov regime-switching. First, we employ the “homothetic robustness” to preserve the performance of robustness for the ALM model, which runs well in precisely modified state variables and performs reasonably if some forms of model misspecification exist. Second, we consider the asset-to-liability ratio instead of the surplus, which ensures that we use relative values instead of absolute values to modify the wealth process. Besides, we use the stochastic dynamic programming method to get some closed-form results and analyze the impacts of parameters on the investment strategy and value function, respectively, by numerical examples.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44383383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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