{"title":"Finite-time horizon, stopper vs. singular-controller games on the half-line","authors":"Andrea Bovo, Tiziano De Angelis","doi":"arxiv-2409.06049","DOIUrl":null,"url":null,"abstract":"We prove existence of a value for two-player zero-sum stopper vs.\nsingular-controller games on finite-time horizon, when the underlying dynamics\nis one-dimensional, diffusive and bound to evolve in $[0,\\infty)$. We show that\nthe value is the maximal solution of a variational inequality with both\nobstacle and gradient constraint and satisfying a Dirichlet boundary condition\nat $[0,T)\\times\\{0\\}$. Moreover, we obtain an optimal strategy for the stopper.\nCompared to the existing literature on this topic, we introduce new\nprobabilistic methods to obtain gradient bounds and equi-continuity for the\nsolutions of penalised partial differential equations (PDE) that approximate\nthe variational inequality.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"101 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06049","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove existence of a value for two-player zero-sum stopper vs.
singular-controller games on finite-time horizon, when the underlying dynamics
is one-dimensional, diffusive and bound to evolve in $[0,\infty)$. We show that
the value is the maximal solution of a variational inequality with both
obstacle and gradient constraint and satisfying a Dirichlet boundary condition
at $[0,T)\times\{0\}$. Moreover, we obtain an optimal strategy for the stopper.
Compared to the existing literature on this topic, we introduce new
probabilistic methods to obtain gradient bounds and equi-continuity for the
solutions of penalised partial differential equations (PDE) that approximate
the variational inequality.