{"title":"积分扩散过程的一个首次通过位置问题","authors":"M. Lefebvre","doi":"10.1017/jpr.2023.19","DOIUrl":null,"url":null,"abstract":"\n Let \n \n \n \n${\\mathrm{d}} X(t) = -Y(t) \\, {\\mathrm{d}} t$\n\n \n , where Y(t) is a one-dimensional diffusion process, and let \n \n \n \n$\\tau(x,y)$\n\n \n be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability \n \n \n \n$p(x,y)\\,:\\!=\\, \\mathbb{P}[X(\\tau(x,y))=0]$\n\n \n is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of \n \n \n \n$\\mathbb{E}[\\tau(x,y)]$\n\n \n and of the moment-generating function of \n \n \n \n$\\tau(x,y)$\n\n \n can also be derived.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A first-passage-place problem for integrated diffusion processes\",\"authors\":\"M. Lefebvre\",\"doi\":\"10.1017/jpr.2023.19\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n Let \\n \\n \\n \\n${\\\\mathrm{d}} X(t) = -Y(t) \\\\, {\\\\mathrm{d}} t$\\n\\n \\n , where Y(t) is a one-dimensional diffusion process, and let \\n \\n \\n \\n$\\\\tau(x,y)$\\n\\n \\n be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability \\n \\n \\n \\n$p(x,y)\\\\,:\\\\!=\\\\, \\\\mathbb{P}[X(\\\\tau(x,y))=0]$\\n\\n \\n is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of \\n \\n \\n \\n$\\\\mathbb{E}[\\\\tau(x,y)]$\\n\\n \\n and of the moment-generating function of \\n \\n \\n \\n$\\\\tau(x,y)$\\n\\n \\n can also be derived.\",\"PeriodicalId\":50256,\"journal\":{\"name\":\"Journal of Applied Probability\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/jpr.2023.19\",\"RegionNum\":4,\"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":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2023.19","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
A first-passage-place problem for integrated diffusion processes
Let
${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$
, where Y(t) is a one-dimensional diffusion process, and let
$\tau(x,y)$
be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability
$p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$
is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of
$\mathbb{E}[\tau(x,y)]$
and of the moment-generating function of
$\tau(x,y)$
can also be derived.
期刊介绍:
Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used.
A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.