Pub Date : 2022-09-15DOI: 10.1080/15326349.2022.2114496
R. Butler
Abstract Asymptotic residue expansions are proposed for inverting probability generating functions (PGFs) and approximating their associated mass and survival functions. The expansions are useful in the wide range of stochastic model applications in which a PGF admits poles in its analytic continuation. The error of such an expansion is a contour integral in the analytic continuation and saddlepoint approximations are developed for such errors using the method of steepest descents. These saddlepoint error estimates attain sufficient accuracy that they can be used to set the order of the expansion so it achieves a specified error. Numerical applications include a success run tutorial example, the discrete ruin model, the Pollaczek-Khintchine formula, and passage times for semi-Markov processes. The residue expansions apply more generally for inverting generating functions which arise in renewal theory and combinatorics and lead to a simple proof of the classic renewal theorem. They extend even further for determining Taylor coefficients of general meromorphic functions.
{"title":"Residue expansions and saddlepoint approximations in stochastic models using the analytic continuation of generating functions","authors":"R. Butler","doi":"10.1080/15326349.2022.2114496","DOIUrl":"https://doi.org/10.1080/15326349.2022.2114496","url":null,"abstract":"Abstract Asymptotic residue expansions are proposed for inverting probability generating functions (PGFs) and approximating their associated mass and survival functions. The expansions are useful in the wide range of stochastic model applications in which a PGF admits poles in its analytic continuation. The error of such an expansion is a contour integral in the analytic continuation and saddlepoint approximations are developed for such errors using the method of steepest descents. These saddlepoint error estimates attain sufficient accuracy that they can be used to set the order of the expansion so it achieves a specified error. Numerical applications include a success run tutorial example, the discrete ruin model, the Pollaczek-Khintchine formula, and passage times for semi-Markov processes. The residue expansions apply more generally for inverting generating functions which arise in renewal theory and combinatorics and lead to a simple proof of the classic renewal theorem. They extend even further for determining Taylor coefficients of general meromorphic functions.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"469 - 501"},"PeriodicalIF":0.7,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47225723","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 : 2022-09-10DOI: 10.1080/15326349.2022.2117195
Junjun Zheng, H. Okamura, T. Dohi
{"title":"Sensitivity analysis for a Markov regenerative software rejuvenation model","authors":"Junjun Zheng, H. Okamura, T. Dohi","doi":"10.1080/15326349.2022.2117195","DOIUrl":"https://doi.org/10.1080/15326349.2022.2117195","url":null,"abstract":"","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43935327","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 : 2022-08-27DOI: 10.1080/15326349.2022.2112225
Bora Çekyay, S. Özekici
Abstract We analyze mean time to failure and availability of systems that perform semi-Markov missions. The mission process is the minimal semi-Markov process associated with a Markov renewal process. Therefore, the successive phases of the mission follow a Markov chain, and the phase durations are generally distributed. The lifetimes of the non-identical components in the system are assumed to be generally distributed and are modeled using intrinsic aging concepts. Moreover, the lifetime parameters of the components and the configuration of the system change depending on the phases of the mission. We characterize the mean time to failure through solving a Poisson equation, and we analyze the system availability assuming that repair duration has a general distribution which is dependent on the phase of the mission during which the failure has occurred and on the deterioration level of the system.
{"title":"MTTF and availability of semi-Markov missions with non-identical generally distributed component lifetimes","authors":"Bora Çekyay, S. Özekici","doi":"10.1080/15326349.2022.2112225","DOIUrl":"https://doi.org/10.1080/15326349.2022.2112225","url":null,"abstract":"Abstract We analyze mean time to failure and availability of systems that perform semi-Markov missions. The mission process is the minimal semi-Markov process associated with a Markov renewal process. Therefore, the successive phases of the mission follow a Markov chain, and the phase durations are generally distributed. The lifetimes of the non-identical components in the system are assumed to be generally distributed and are modeled using intrinsic aging concepts. Moreover, the lifetime parameters of the components and the configuration of the system change depending on the phases of the mission. We characterize the mean time to failure through solving a Poisson equation, and we analyze the system availability assuming that repair duration has a general distribution which is dependent on the phase of the mission during which the failure has occurred and on the deterioration level of the system.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"414 - 447"},"PeriodicalIF":0.7,"publicationDate":"2022-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46798700","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 : 2022-08-22DOI: 10.1080/15326349.2022.2107666
S. Kayal, Raju Bhakta, N. Balakrishnan
Abstract Finite mixture (FM) models have found key applications in many fields. Recently, some discussions have been made on comparing finite mixture models. In this paper, we discuss stochastic comparison of two FM models with respect to usual stochastic order when the mixture components have a general family of distributions. This problem has been studied when there is heterogeneity in one parameter (i.e., the distributional parameter), as well as when there is heterogeneity in two parameters (i.e., the distributional parameter and the mixing proportions). The sufficient conditions considered are based on p-larger order and reciprocally majorization order. Several examples have been provided to illustrate the established results.
{"title":"Some results on stochastic comparisons of two finite mixture models with general components","authors":"S. Kayal, Raju Bhakta, N. Balakrishnan","doi":"10.1080/15326349.2022.2107666","DOIUrl":"https://doi.org/10.1080/15326349.2022.2107666","url":null,"abstract":"Abstract Finite mixture (FM) models have found key applications in many fields. Recently, some discussions have been made on comparing finite mixture models. In this paper, we discuss stochastic comparison of two FM models with respect to usual stochastic order when the mixture components have a general family of distributions. This problem has been studied when there is heterogeneity in one parameter (i.e., the distributional parameter), as well as when there is heterogeneity in two parameters (i.e., the distributional parameter and the mixing proportions). The sufficient conditions considered are based on p-larger order and reciprocally majorization order. Several examples have been provided to illustrate the established results.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"363 - 382"},"PeriodicalIF":0.7,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48858509","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 : 2022-08-19DOI: 10.1080/15326349.2022.2108452
V. Ejov, J. Filar, Zhihao Qiao
Abstract We consider the problem of parametric sensitivity of a particular characterization of risk, with respect to a threshold parameter Such threshold risk is modeled as the probability of a perturbed function of a random variable falling below 0. We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as δ approaches them. Under weak conditions, we characterize candidates for risk critical points as zeroes of either the discriminant of a relevant perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in δ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as” hidden equations of risk critical thresholds”.
{"title":"Hidden equations of risk critical thresholds","authors":"V. Ejov, J. Filar, Zhihao Qiao","doi":"10.1080/15326349.2022.2108452","DOIUrl":"https://doi.org/10.1080/15326349.2022.2108452","url":null,"abstract":"Abstract We consider the problem of parametric sensitivity of a particular characterization of risk, with respect to a threshold parameter Such threshold risk is modeled as the probability of a perturbed function of a random variable falling below 0. We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as δ approaches them. Under weak conditions, we characterize candidates for risk critical points as zeroes of either the discriminant of a relevant perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in δ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as” hidden equations of risk critical thresholds”.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"383 - 413"},"PeriodicalIF":0.7,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48254482","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 : 2022-08-15DOI: 10.1080/15326349.2022.2100423
Jörn Sass, Dorothee Westphal, R. Wunderlich
Abstract In this paper we study a financial market in which stock returns depend on an unobservable Gaussian drift process. Investors obtain information on that drift from return observations and discrete-time expert opinions as an external source of information. Estimates of the hidden drift process are based on filtering techniques. Our focus is the case of high-frequency experts periodically providing their views on the drift with variances growing linearly with the arrival frequency. The latter condition guarantees that the delivered information per time is limited. The asymptotic behavior of the filter as the arrival frequency tends to infinity is described by limit theorems. These state that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. We apply these diffusion approximations of the filter for deriving simplified approximate solutions of utility maximization problems with logarithmic and power utility.
{"title":"Diffusion approximations for periodically arriving expert opinions in a financial market with Gaussian drift","authors":"Jörn Sass, Dorothee Westphal, R. Wunderlich","doi":"10.1080/15326349.2022.2100423","DOIUrl":"https://doi.org/10.1080/15326349.2022.2100423","url":null,"abstract":"Abstract In this paper we study a financial market in which stock returns depend on an unobservable Gaussian drift process. Investors obtain information on that drift from return observations and discrete-time expert opinions as an external source of information. Estimates of the hidden drift process are based on filtering techniques. Our focus is the case of high-frequency experts periodically providing their views on the drift with variances growing linearly with the arrival frequency. The latter condition guarantees that the delivered information per time is limited. The asymptotic behavior of the filter as the arrival frequency tends to infinity is described by limit theorems. These state that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. We apply these diffusion approximations of the filter for deriving simplified approximate solutions of utility maximization problems with logarithmic and power utility.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"39 1","pages":"323 - 362"},"PeriodicalIF":0.7,"publicationDate":"2022-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46582844","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 : 2022-08-09DOI: 10.1080/15326349.2022.2107667
S. Inoue, S. Yamada, T. Fujiwara
{"title":"Economic shipping policies for assuring safety integrity level of E/E/PE safety-related software","authors":"S. Inoue, S. Yamada, T. Fujiwara","doi":"10.1080/15326349.2022.2107667","DOIUrl":"https://doi.org/10.1080/15326349.2022.2107667","url":null,"abstract":"","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44822469","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 : 2022-07-04DOI: 10.1080/15326349.2023.2185257
Martin Bladt, Clara Brimnes Gardner
In this paper we introduce a bivariate distribution on $mathbb{R}_{+} times mathbb{N}$ arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on $mathbb{R}_{+} times mathbb{N}$ and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.
{"title":"Joint discrete and continuous matrix distribution modeling","authors":"Martin Bladt, Clara Brimnes Gardner","doi":"10.1080/15326349.2023.2185257","DOIUrl":"https://doi.org/10.1080/15326349.2023.2185257","url":null,"abstract":"In this paper we introduce a bivariate distribution on $mathbb{R}_{+} times mathbb{N}$ arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on $mathbb{R}_{+} times mathbb{N}$ and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41511202","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 : 2022-06-07DOI: 10.1080/15326349.2022.2081206
M. Telek
Abstract The paper investigates the relation of random clock based and numerical inverse Laplace transformation based transient analysis of Continuous time Markov chains (CTMCs) and Markov fluid models (MFMs) and proves that these methods are identical. This identity leads to new intuitive understanding about the analysis approaches.
{"title":"Transient analysis of Markov modulated processes with Erlangization, ME-fication and inverse Laplace transformation","authors":"M. Telek","doi":"10.1080/15326349.2022.2081206","DOIUrl":"https://doi.org/10.1080/15326349.2022.2081206","url":null,"abstract":"Abstract The paper investigates the relation of random clock based and numerical inverse Laplace transformation based transient analysis of Continuous time Markov chains (CTMCs) and Markov fluid models (MFMs) and proves that these methods are identical. This identity leads to new intuitive understanding about the analysis approaches.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"38 1","pages":"638 - 664"},"PeriodicalIF":0.7,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46544739","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 : 2022-06-07DOI: 10.1080/15326349.2022.2080709
Arash Fahim, Lingjiong Zhu
Abstract
The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for a dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividend problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.
{"title":"Asymptotic analysis for optimal dividends in a dual risk model","authors":"Arash Fahim, Lingjiong Zhu","doi":"10.1080/15326349.2022.2080709","DOIUrl":"https://doi.org/10.1080/15326349.2022.2080709","url":null,"abstract":"<p><b>Abstract</b></p><p>The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for a dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividend problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.</p>","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536192","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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