Advances in Applied Probability最新文献

Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains 近密度依赖马尔可夫链的一维受限朗文近似的误差边界

Advances in Applied Probability Pub Date : 2024-07-26 DOI: 10.1017/apr.2024.25

Felipe A. Campos, Ruth J. Williams

{"title":"Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains","authors":"Felipe A. Campos, Ruth J. Williams","doi":"10.1017/apr.2024.25","DOIUrl":"https://doi.org/10.1017/apr.2024.25","url":null,"abstract":"\u0000 Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob.29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul.17, 2019).\u0000 In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.","PeriodicalId":502238,"journal":{"name":"Advances in Applied Probability","volume":"120 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141801834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}