{"title":"2. Presentation of the Main Results","authors":"","doi":"10.1515/9780691193717-003","DOIUrl":"https://doi.org/10.1515/9780691193717-003","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126650596","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}
{"title":"Index","authors":"","doi":"10.1515/9780691193717-010","DOIUrl":"https://doi.org/10.1515/9780691193717-010","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117010644","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}
{"title":"Frontmatter","authors":"","doi":"10.1515/9780691193717-fm","DOIUrl":"https://doi.org/10.1515/9780691193717-fm","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129816487","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}
{"title":"6. Convergence of the Nash System","authors":"","doi":"10.1515/9780691193717-007","DOIUrl":"https://doi.org/10.1515/9780691193717-007","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125376795","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}
{"title":"5. The Second-Order Master Equation","authors":"","doi":"10.1515/9780691193717-006","DOIUrl":"https://doi.org/10.1515/9780691193717-006","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115906515","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}
{"title":"3. A Starter: The First-Order Master Equation","authors":"","doi":"10.1515/9780691193717-004","DOIUrl":"https://doi.org/10.1515/9780691193717-004","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121882801","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}
{"title":"The Second-Order Master Equation","authors":"P. Cardaliaguet, F. Delarue, J. Lasry, P. Lions","doi":"10.2307/j.ctvckq7qf.8","DOIUrl":"https://doi.org/10.2307/j.ctvckq7qf.8","url":null,"abstract":"This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130229810","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}
{"title":"Presentation of the Main Results","authors":"P. Cardaliaguet, F. Delarue, J. Lasry, P. Lions","doi":"10.2307/j.ctvckq7qf.5","DOIUrl":"https://doi.org/10.2307/j.ctvckq7qf.5","url":null,"abstract":"This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126014578","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}
{"title":"Convergence of the Nash System","authors":"P. Cardaliaguet, F. Delarue, J. Lasry, P. Lions","doi":"10.2307/j.ctvckq7qf.9","DOIUrl":"https://doi.org/10.2307/j.ctvckq7qf.9","url":null,"abstract":"This chapter talks about addressing the convergence problem, which is devoted to the convergence of the Nash system. It contains several results on the differential calculus on the space of probability measures together with an Itô's formula for functionals of a process taking values in the space of probability measures. For simplicity, most of the analysis provided in the chapter is on the torus, but the method is robust enough to accommodate the nonperiodic setting. The chapter also shows that monotonicity plays no role in the proofs of certain theorems. Basically, only the global Lipschitz properties of H and DpH, together with the nondegeneracy of the diffusions and the various bounds obtained for the solution of the master equation and its derivatives matter.","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127788331","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}
{"title":"Index","authors":"","doi":"10.2307/j.ctvckq7qf.12","DOIUrl":"https://doi.org/10.2307/j.ctvckq7qf.12","url":null,"abstract":"","PeriodicalId":306589,"journal":{"name":"The Master Equation and the Convergence Problem in Mean Field Games","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126438232","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}