{"title":"Counting excursions: symmetries, knock-ins and\nnon-linear formula for Itô--McKean diffusions","authors":"Maciej Wiśniewolski","doi":"10.30757/ALEA.V18-16","DOIUrl":null,"url":null,"abstract":"Excursion theory is revisited on the ground of Itô–McKean diffusions. There are raised questions about symmetries, knock-in processes, excursion local time and the non-linear version of the master formula of excursions. The questions are answered due to introducing the counting excursion technique. The technique is a synthesis of straddling at time approach, the classical, potential in spirit approach, and the theory of convolution algebra of locally integrable functions, generalized later in this work for the convolutions of σ–finite measures. Some examples are presented, including the famous problem of expressing the density of first hitting time of Ornstein-Uhlenbeck process in terms of elementary functions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"29 1","pages":"349"},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/ALEA.V18-16","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Excursion theory is revisited on the ground of Itô–McKean diffusions. There are raised questions about symmetries, knock-in processes, excursion local time and the non-linear version of the master formula of excursions. The questions are answered due to introducing the counting excursion technique. The technique is a synthesis of straddling at time approach, the classical, potential in spirit approach, and the theory of convolution algebra of locally integrable functions, generalized later in this work for the convolutions of σ–finite measures. Some examples are presented, including the famous problem of expressing the density of first hitting time of Ornstein-Uhlenbeck process in terms of elementary functions.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.