Pub Date : 2022-02-28DOI: 10.1080/15326349.2022.2036192
Lei Dang, Xamxinur Abdurahman, Z. Teng
Abstract Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations.
{"title":"The threshold dynamics of a stochastic two-patch brucellosis model","authors":"Lei Dang, Xamxinur Abdurahman, Z. Teng","doi":"10.1080/15326349.2022.2036192","DOIUrl":"https://doi.org/10.1080/15326349.2022.2036192","url":null,"abstract":"Abstract Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"38 1","pages":"331 - 364"},"PeriodicalIF":0.7,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42813706","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 : 2022-02-23DOI: 10.1080/15326349.2022.2040365
Chunmao Huang, Chen Wang, Xiaoqiang Wang
Abstract We consider a supercritical discrete-time branching process with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios
{"title":"Moments and asymptotic properties for supercritical branching processes with immigration in random environments","authors":"Chunmao Huang, Chen Wang, Xiaoqiang Wang","doi":"10.1080/15326349.2022.2040365","DOIUrl":"https://doi.org/10.1080/15326349.2022.2040365","url":null,"abstract":"Abstract We consider a supercritical discrete-time branching process with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"21 - 40"},"PeriodicalIF":0.7,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43893984","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 : 2022-02-21DOI: 10.1080/15326349.2022.2033628
A. Imomov, E. E. Tukhtaev
Abstract We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.
{"title":"On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance","authors":"A. Imomov, E. E. Tukhtaev","doi":"10.1080/15326349.2022.2033628","DOIUrl":"https://doi.org/10.1080/15326349.2022.2033628","url":null,"abstract":"Abstract We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"118 - 140"},"PeriodicalIF":0.7,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42335913","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 : 2022-02-21DOI: 10.1080/15326349.2021.2022496
H. Jiang, N. Gibson, Y. Chen
Abstract This paper considers the revenue maximization problem for a hydropower company. The company can generate excess electricity by releasing water from a reservoir and then sell it to the energy market. On the other hand, the company has an obligation to keep the reservoir level above a pre-determined level, which may require the company to purchase electricity in order to fulfill the customers’ power demand. The electricity price and reservoir level are both represented by diffusion processes. We refer to a one-factor diffusion model for electricity price, which is known to fit the data well. After applying Bellman dynamic programming principle, we derive the associated state-constrained Hamilton-Jacobi-Bellman (HJB) equation to characterize the value function. Then we prove that the value function is the viscosity solution of the state-constrained HJB equation and it is unique in this constrained optimization problem.
{"title":"A stochastic model for the optimal allocation of hydropower flexibility in renewable energy markets","authors":"H. Jiang, N. Gibson, Y. Chen","doi":"10.1080/15326349.2021.2022496","DOIUrl":"https://doi.org/10.1080/15326349.2021.2022496","url":null,"abstract":"Abstract This paper considers the revenue maximization problem for a hydropower company. The company can generate excess electricity by releasing water from a reservoir and then sell it to the energy market. On the other hand, the company has an obligation to keep the reservoir level above a pre-determined level, which may require the company to purchase electricity in order to fulfill the customers’ power demand. The electricity price and reservoir level are both represented by diffusion processes. We refer to a one-factor diffusion model for electricity price, which is known to fit the data well. After applying Bellman dynamic programming principle, we derive the associated state-constrained Hamilton-Jacobi-Bellman (HJB) equation to characterize the value function. Then we prove that the value function is the viscosity solution of the state-constrained HJB equation and it is unique in this constrained optimization problem.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"38 1","pages":"288 - 307"},"PeriodicalIF":0.7,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45242335","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 : 2022-02-07DOI: 10.1080/15326349.2022.2026790
S. Ankirchner, Christophette Blanchet-Scalliet, Diana Dorobantu, Laura Gay
Abstract We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.
{"title":"First passage time density of an Ornstein–Uhlenbeck process with broken drift","authors":"S. Ankirchner, Christophette Blanchet-Scalliet, Diana Dorobantu, Laura Gay","doi":"10.1080/15326349.2022.2026790","DOIUrl":"https://doi.org/10.1080/15326349.2022.2026790","url":null,"abstract":"Abstract We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"38 1","pages":"308 - 329"},"PeriodicalIF":0.7,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47347357","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 : 2022-01-21DOI: 10.1080/15326349.2021.2016446
M. Slavtchova-Bojkova, O. Hyrien, N. Yanev
Abstract We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.
{"title":"Poisson random measures and supercritical multitype Markov branching processes","authors":"M. Slavtchova-Bojkova, O. Hyrien, N. Yanev","doi":"10.1080/15326349.2021.2016446","DOIUrl":"https://doi.org/10.1080/15326349.2021.2016446","url":null,"abstract":"Abstract We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"141 - 160"},"PeriodicalIF":0.7,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49099878","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 : 2022-01-11DOI: 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":"38 1","pages":"250 - 267"},"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":"38 1","pages":"268 - 287"},"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":" ","pages":""},"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":"39 1","pages":"5 - 20"},"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}
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