{"title":"不连续小噪声非线性滤波的大偏差估计","authors":"Hongjiang Qian , Yanzhao Cao , George Yin","doi":"10.1016/j.spa.2025.104662","DOIUrl":null,"url":null,"abstract":"<div><div>This paper develops large deviation estimates for nonlinear filtering with discontinuity in the drift of the signal and small noise intensities in both the signal and the observations. A variational approach related to Mortensen’s optimization problem is utilized in our analysis. The discontinuity of the drift in the signal naturally arises in many applications, including modeling communication channels with a “hard limiter”. Our results extend the work of Reddy et al. (2022), in which smooth functions were used. To address the discontinuous in the drift of the signal, relaxed controls are used to study the asymptotic fraction of time the controlled signals spend in each half-space divided by the discontinuity hyperplane. Large deviation estimates are established by the weak convergence method using the stochastic control representation for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur–Striebel formula.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104662"},"PeriodicalIF":1.1000,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Large deviation estimates for nonlinear filtering with discontinuity and small noise\",\"authors\":\"Hongjiang Qian , Yanzhao Cao , George Yin\",\"doi\":\"10.1016/j.spa.2025.104662\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper develops large deviation estimates for nonlinear filtering with discontinuity in the drift of the signal and small noise intensities in both the signal and the observations. A variational approach related to Mortensen’s optimization problem is utilized in our analysis. The discontinuity of the drift in the signal naturally arises in many applications, including modeling communication channels with a “hard limiter”. Our results extend the work of Reddy et al. (2022), in which smooth functions were used. To address the discontinuous in the drift of the signal, relaxed controls are used to study the asymptotic fraction of time the controlled signals spend in each half-space divided by the discontinuity hyperplane. Large deviation estimates are established by the weak convergence method using the stochastic control representation for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur–Striebel formula.</div></div>\",\"PeriodicalId\":51160,\"journal\":{\"name\":\"Stochastic Processes and their Applications\",\"volume\":\"187 \",\"pages\":\"Article 104662\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2025-04-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Stochastic Processes and their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304414925001036\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414925001036","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Large deviation estimates for nonlinear filtering with discontinuity and small noise
This paper develops large deviation estimates for nonlinear filtering with discontinuity in the drift of the signal and small noise intensities in both the signal and the observations. A variational approach related to Mortensen’s optimization problem is utilized in our analysis. The discontinuity of the drift in the signal naturally arises in many applications, including modeling communication channels with a “hard limiter”. Our results extend the work of Reddy et al. (2022), in which smooth functions were used. To address the discontinuous in the drift of the signal, relaxed controls are used to study the asymptotic fraction of time the controlled signals spend in each half-space divided by the discontinuity hyperplane. Large deviation estimates are established by the weak convergence method using the stochastic control representation for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur–Striebel formula.
期刊介绍:
Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests.
Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.