PDE for the joint law of the pair of a continuous diffusion and its running maximum

Pub Date : 2023-01-06 DOI:10.1017/apr.2022.76

L. Coutin, M. Pontier

{"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 Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any \n\t \n\t\t\n\t\t\n$t>0,$\n\n\t \n\t the density (with respect to the \n\t \n\t\t\n\t\t\n$(d+1)$\n\n\t \n\t -dimensional Lebesgue measure) of the pair \n\t \n\t\t\n\t\t\n$\\big(M_t,X_t\\big)$\n\n\t \n\t is a weak solution of a Fokker–Planck partial differential equation on the closed set \n\t \n\t\t\n\t\t\n$\\big\\{(m,x)\\in \\mathbb{R}^{d+1},\\,{m\\geq x^1}\\big\\},$\n\n\t \n\t using an integral expansion of this density.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/apr.2022.76","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}