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Stochastic ordering results on the duration of the gambler’s ruin game 赌徒破产博弈持续时间的随机排序结果
4区 数学
Journal of Applied Probability Pub Date : 2023-10-06 DOI: 10.1017/jpr.2023.62
Shoou-Ren Hsiau, Yi-Ching Yao
{"title":"Stochastic ordering results on the duration of the gambler’s ruin game","authors":"Shoou-Ren Hsiau, Yi-Ching Yao","doi":"10.1017/jpr.2023.62","DOIUrl":"https://doi.org/10.1017/jpr.2023.62","url":null,"abstract":"Abstract In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0<zleq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p in [0,{frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} leq z<a$ , the family of distributions of the duration indexed by $p in [{frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p in [0,1]$ by $p={frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351786","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}
引用次数: 0
Optimal stopping methodology for the secretary problem with random queries 随机查询秘书问题的最优停止方法
4区 数学
Journal of Applied Probability Pub Date : 2023-10-02 DOI: 10.1017/jpr.2023.61
George V. Moustakides, Xujun Liu, Olgica Milenkovic
{"title":"Optimal stopping methodology for the secretary problem with random queries","authors":"George V. Moustakides, Xujun Liu, Olgica Milenkovic","doi":"10.1017/jpr.2023.61","DOIUrl":"https://doi.org/10.1017/jpr.2023.61","url":null,"abstract":"Abstract Candidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, a decision mechanism must be developed that selects or dismisses the current candidate in an effort to maximize the chance of selecting the best. This classical version of the ‘secretary problem’ has been studied in depth, mostly using combinatorial approaches, along with numerous other variants. We consider a particular new version where, during reviewing, it is possible to query an external expert to improve the probability of making the correct decision. Unlike existing formulations, we consider experts that are not necessarily infallible and may provide suggestions that can be faulty. For the solution of our problem we adopt a probabilistic methodology and view the querying times as consecutive stopping times which we optimize with the help of optimal stopping theory. For each querying time we must also design a mechanism to decide whether or not we should terminate the search at the querying time. This decision is straightforward under the usual assumption of infallible experts, but when experts are faulty it has a far more intricate structure.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790097","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}
引用次数: 2
Stochastic differential equation approximations of generative adversarial network training and its long-run behavior 生成对抗网络训练的随机微分方程逼近及其长期行为
4区 数学
Journal of Applied Probability Pub Date : 2023-10-02 DOI: 10.1017/jpr.2023.57
Haoyang Cao, Xin Guo
{"title":"Stochastic differential equation approximations of generative adversarial network training and its long-run behavior","authors":"Haoyang Cao, Xin Guo","doi":"10.1017/jpr.2023.57","DOIUrl":"https://doi.org/10.1017/jpr.2023.57","url":null,"abstract":"Abstract This paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135831357","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}
引用次数: 0
R-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices 最近邻矩阵上Gibbs测度的r -正性和零温度极限的存在性
4区 数学
Journal of Applied Probability Pub Date : 2023-09-25 DOI: 10.1017/jpr.2023.59
Jorge Littin Curinao, Gerardo Corredor Rincón
{"title":"<i>R</i>-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices","authors":"Jorge Littin Curinao, Gerardo Corredor Rincón","doi":"10.1017/jpr.2023.59","DOIUrl":"https://doi.org/10.1017/jpr.2023.59","url":null,"abstract":"Abstract We study the $R_beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(mu_{beta}(!cdot!))_{beta geq 0}$ at inverse temperature $beta$ associated to a family of nearest-neighbor matrices $(Q_{beta})_{beta geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $beta$ , and (ii) the existence of weak limits as $beta to infty$ . Some application examples are revised to put in context the main results of this work.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135816835","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}
引用次数: 0
Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes 状态切换过程转移速率矩阵摄动下的比较定理与稳定性
4区 数学
Journal of Applied Probability Pub Date : 2023-09-14 DOI: 10.1017/jpr.2023.54
Jinghai Shao
{"title":"Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes","authors":"Jinghai Shao","doi":"10.1017/jpr.2023.54","DOIUrl":"https://doi.org/10.1017/jpr.2023.54","url":null,"abstract":"Abstract A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911852","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}
引用次数: 0
Reliability analyses of linear two-dimensional consecutive k-type systems 线性二维连续k型系统的可靠性分析
IF 1 4区 数学
Journal of Applied Probability Pub Date : 2023-08-14 DOI: 10.1017/jpr.2023.51
He Yi, N. Balakrishnan, Xiang Li
{"title":"Reliability analyses of linear two-dimensional consecutive k-type systems","authors":"He Yi, N. Balakrishnan, Xiang Li","doi":"10.1017/jpr.2023.51","DOIUrl":"https://doi.org/10.1017/jpr.2023.51","url":null,"abstract":"\u0000 In this paper, several linear two-dimensional consecutive k-type systems are studied, which include the linear connected-(k, r)-out-of-\u0000 \u0000 \u0000 \u0000$(m,n)colon! F$\u0000\u0000 \u0000 system and the linear l-connected-(k, r)-out-of-\u0000 \u0000 \u0000 \u0000$(m,n)colon! F$\u0000\u0000 \u0000 system without/with overlapping. Reliabilities of these systems are studied via the finite Markov chain imbedding approach (FMCIA) in a novel way. Some numerical examples are provided to illustrate the theoretical results established here and also to demonstrate the efficiency of the developed method. Finally, some possible applications and generalizations of the developed results are pointed out.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43227810","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}
引用次数: 1
JPR volume 60 issue 3 Cover and Back matter JPR第60卷第3期封面和封底
IF 1 4区 数学
Journal of Applied Probability Pub Date : 2023-08-07 DOI: 10.1017/jpr.2023.42
{"title":"JPR volume 60 issue 3 Cover and Back matter","authors":"","doi":"10.1017/jpr.2023.42","DOIUrl":"https://doi.org/10.1017/jpr.2023.42","url":null,"abstract":"","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"b1 - b2"},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43006388","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}
引用次数: 0
JPR volume 60 issue 3 Cover and Front matter JPR第60卷第3期封面和封面
IF 1 4区 数学
Journal of Applied Probability Pub Date : 2023-08-07 DOI: 10.1017/jpr.2023.41
{"title":"JPR volume 60 issue 3 Cover and Front matter","authors":"","doi":"10.1017/jpr.2023.41","DOIUrl":"https://doi.org/10.1017/jpr.2023.41","url":null,"abstract":"","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":"f1 - f2"},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47078496","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}
引用次数: 0
Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes 具有随机分割Hawkes到达过程的并行单服务器队列的重流量限制
IF 1 4区 数学
Journal of Applied Probability Pub Date : 2023-08-07 DOI: 10.1017/jpr.2023.50
Bo Li, G. Pang
{"title":"Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes","authors":"Bo Li, G. Pang","doi":"10.1017/jpr.2023.50","DOIUrl":"https://doi.org/10.1017/jpr.2023.50","url":null,"abstract":"\u0000 We consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45371790","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}
引用次数: 0
On the joint survival probability of two collaborating firms 关于两个合作企业的共同生存概率
IF 1 4区 数学
Journal of Applied Probability Pub Date : 2023-08-01 DOI: 10.1017/jpr.2023.46
S. Ankirchner, R. Hesse, Maike Klein
{"title":"On the joint survival probability of two collaborating firms","authors":"S. Ankirchner, R. Hesse, Maike Klein","doi":"10.1017/jpr.2023.46","DOIUrl":"https://doi.org/10.1017/jpr.2023.46","url":null,"abstract":"\u0000 We consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42352603","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}
引用次数: 1
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