{"title":"PDE为连续扩散对及其运行最大值的联合律","authors":"L. Coutin, M. Pontier","doi":"10.1017/apr.2022.76","DOIUrl":null,"url":null,"abstract":"\n\t <jats:p>Let <jats:italic>X</jats:italic> be a <jats:italic>d</jats:italic>-dimensional diffusion and <jats:italic>M</jats:italic> the running supremum of its first component. In this paper, we show that for any <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline1.png\" />\n\t\t<jats:tex-math>\n$t>0,$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> the density (with respect to the <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline2.png\" />\n\t\t<jats:tex-math>\n$(d+1)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>-dimensional Lebesgue measure) of the pair <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline3.png\" />\n\t\t<jats:tex-math>\n$\\big(M_t,X_t\\big)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> is a weak solution of a Fokker–Planck partial differential equation on the closed set <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0001867822000763_inline4.png\" />\n\t\t<jats:tex-math>\n$\\big\\{(m,x)\\in \\mathbb{R}^{d+1},\\,{m\\geq x^1}\\big\\},$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> using an integral expansion of this density.</jats:p>","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"PDE for the joint law of the pair of a continuous diffusion and its running maximum\",\"authors\":\"L. Coutin, M. Pontier\",\"doi\":\"10.1017/apr.2022.76\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n\\t <jats:p>Let <jats:italic>X</jats:italic> be a <jats:italic>d</jats:italic>-dimensional diffusion and <jats:italic>M</jats:italic> the running supremum of its first component. In this paper, we show that for any <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0001867822000763_inline1.png\\\" />\\n\\t\\t<jats:tex-math>\\n$t>0,$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula> the density (with respect to the <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0001867822000763_inline2.png\\\" />\\n\\t\\t<jats:tex-math>\\n$(d+1)$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula>-dimensional Lebesgue measure) of the pair <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0001867822000763_inline3.png\\\" />\\n\\t\\t<jats:tex-math>\\n$\\\\big(M_t,X_t\\\\big)$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula> is a weak solution of a Fokker–Planck partial differential equation on the closed set <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0001867822000763_inline4.png\\\" />\\n\\t\\t<jats:tex-math>\\n$\\\\big\\\\{(m,x)\\\\in \\\\mathbb{R}^{d+1},\\\\,{m\\\\geq x^1}\\\\big\\\\},$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula> using an integral expansion of this density.</jats:p>\",\"PeriodicalId\":53160,\"journal\":{\"name\":\"Advances in Applied Probability\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-01-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/apr.2022.76\",\"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":"Advances in Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/apr.2022.76","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
PDE for the joint law of the pair of a continuous diffusion and its running maximum
Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any
$t>0,$
the density (with respect to the
$(d+1)$
-dimensional Lebesgue measure) of the pair
$\big(M_t,X_t\big)$
is a weak solution of a Fokker–Planck partial differential equation on the closed set
$\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$
using an integral expansion of this density.
期刊介绍：
The Advances in Applied Probability has been published by the Applied Probability Trust for over four decades, and is a companion publication to the Journal of Applied Probability. It contains mathematical and scientific papers of interest to applied probabilists, with emphasis on applications in a broad spectrum of disciplines, including the biosciences, 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.
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