{"title":"参数相关粗糙 SDEs 及其在粗糙 PDEs 中的应用","authors":"Fabio Bugini, Peter K. Friz, Wilhelm Stannat","doi":"arxiv-2409.11330","DOIUrl":null,"url":null,"abstract":"In this paper we generalize Krylov's theory on parameter-dependent stochastic\ndifferential equations to the framework of rough stochastic differential\nequations (rough SDEs), as initially introduced by Friz, Hocquet and L\\^e. We\nconsider a stochastic equation of the form $$ dX_t^\\zeta = b_t(\\zeta,X_t^\\zeta)\n\\ dt + \\sigma_t(\\zeta,X_t^\\zeta) \\ dB_t + \\beta_t (\\zeta,X_t^\\zeta)\nd\\mathbf{W}_t,$$ where $\\zeta$ is a parameter, $B$ denotes a Brownian motion\nand $\\mathbf{W}$ is a deterministic H\\\"older rough path. We investigate the\nconditions under which the solution $X$ exhibits continuity and/or\ndifferentiability with respect to the parameter $\\zeta$ in the\n$\\mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class\nof rough partial differential equations (rough PDEs) of the form $$-du_t = L_t\nu_t dt + \\Gamma_t u_t d\\mathbf{W}_t, \\quad u_T =g.$$ We show that the solution\nadmits a Feynman--Kac type representation in terms of the solution of an\nappropriate rough SDE, where the initial time and the initial state play the\nrole of parameters.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parameter dependent rough SDEs with applications to rough PDEs\",\"authors\":\"Fabio Bugini, Peter K. Friz, Wilhelm Stannat\",\"doi\":\"arxiv-2409.11330\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we generalize Krylov's theory on parameter-dependent stochastic\\ndifferential equations to the framework of rough stochastic differential\\nequations (rough SDEs), as initially introduced by Friz, Hocquet and L\\\\^e. We\\nconsider a stochastic equation of the form $$ dX_t^\\\\zeta = b_t(\\\\zeta,X_t^\\\\zeta)\\n\\\\ dt + \\\\sigma_t(\\\\zeta,X_t^\\\\zeta) \\\\ dB_t + \\\\beta_t (\\\\zeta,X_t^\\\\zeta)\\nd\\\\mathbf{W}_t,$$ where $\\\\zeta$ is a parameter, $B$ denotes a Brownian motion\\nand $\\\\mathbf{W}$ is a deterministic H\\\\\\\"older rough path. We investigate the\\nconditions under which the solution $X$ exhibits continuity and/or\\ndifferentiability with respect to the parameter $\\\\zeta$ in the\\n$\\\\mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class\\nof rough partial differential equations (rough PDEs) of the form $$-du_t = L_t\\nu_t dt + \\\\Gamma_t u_t d\\\\mathbf{W}_t, \\\\quad u_T =g.$$ We show that the solution\\nadmits a Feynman--Kac type representation in terms of the solution of an\\nappropriate rough SDE, where the initial time and the initial state play the\\nrole of parameters.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"2 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"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-2409.11330\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11330","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Parameter dependent rough SDEs with applications to rough PDEs
In this paper we generalize Krylov's theory on parameter-dependent stochastic
differential equations to the framework of rough stochastic differential
equations (rough SDEs), as initially introduced by Friz, Hocquet and L\^e. We
consider a stochastic equation of the form $$ dX_t^\zeta = b_t(\zeta,X_t^\zeta)
\ dt + \sigma_t(\zeta,X_t^\zeta) \ dB_t + \beta_t (\zeta,X_t^\zeta)
d\mathbf{W}_t,$$ where $\zeta$ is a parameter, $B$ denotes a Brownian motion
and $\mathbf{W}$ is a deterministic H\"older rough path. We investigate the
conditions under which the solution $X$ exhibits continuity and/or
differentiability with respect to the parameter $\zeta$ in the
$\mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class
of rough partial differential equations (rough PDEs) of the form $$-du_t = L_t
u_t dt + \Gamma_t u_t d\mathbf{W}_t, \quad u_T =g.$$ We show that the solution
admits a Feynman--Kac type representation in terms of the solution of an
appropriate rough SDE, where the initial time and the initial state play the
role of parameters.