We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without reflection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, and we exemplify a situation where this observation allows to derive results for quadratic delayed equations with non-bounded terminal conditions in multi-dimension.
{"title":"BSDEs on Finite and Infinite Horizon with Time-Delayed Generators","authors":"Peng Luo, Ludovic Tangpi","doi":"10.31390/COSA.12.1.05","DOIUrl":"https://doi.org/10.31390/COSA.12.1.05","url":null,"abstract":"We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without reflection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, and we exemplify a situation where this observation allows to derive results for quadratic delayed equations with non-bounded terminal conditions in multi-dimension.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69689626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An analog of the Trotter formula for the Arratia flow is presented. Perturbations of the Brownian web by mappings associated with an ordinary differential equation with a smooth right part are considered and proved to be convergent exclusively in the weak sense. The flow obtained as a limit is the Arratia flow with drift.
{"title":"Arratia Flow with Drift and Trotter Formula for Brownian Web","authors":"A. Dorogovtsev, M. B. Vovchanskii","doi":"10.31390/cosa.12.1.07","DOIUrl":"https://doi.org/10.31390/cosa.12.1.07","url":null,"abstract":"An analog of the Trotter formula for the Arratia flow is presented. Perturbations of the Brownian web by mappings associated with an ordinary differential equation with a smooth right part are considered and proved to be convergent exclusively in the weak sense. The flow obtained as a limit is the Arratia flow with drift.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69689757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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