{"title":"A Two-Person Zero-Sum Game Approach for a Retirement Decision with Borrowing Constraints","authors":"Junkee Jeon, Hyeng Keun Koo, Minsuk Kwak","doi":"10.1137/22m1528124","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 883-930, September 2024. <br/> Abstract. We study an optimal consumption, investment, and retirement decision of an economic agent with borrowing constraints under a general class of utility functions. We transform the problem into a dual two-person zero-sum game, which involves two players: a stopper who is a maximizer and chooses a stopping time and a controller who is a minimizer and chooses a nonincreasing process. We derive the Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) of a max-min type from the dual two-person zero-sum game. We provide a solution to the HJBQVI and verify that the solution to the HJBQVI is the value of the dual two-person zero-sum game. We establish the duality result which allows us to derive the optimal strategies and value function of the primal problem from those of the dual problem. We provide examples for a class of utility functions.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Financial Mathematics","FirstCategoryId":"96","ListUrlMain":"https://doi.org/10.1137/22m1528124","RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"BUSINESS, FINANCE","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 883-930, September 2024. Abstract. We study an optimal consumption, investment, and retirement decision of an economic agent with borrowing constraints under a general class of utility functions. We transform the problem into a dual two-person zero-sum game, which involves two players: a stopper who is a maximizer and chooses a stopping time and a controller who is a minimizer and chooses a nonincreasing process. We derive the Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) of a max-min type from the dual two-person zero-sum game. We provide a solution to the HJBQVI and verify that the solution to the HJBQVI is the value of the dual two-person zero-sum game. We establish the duality result which allows us to derive the optimal strategies and value function of the primal problem from those of the dual problem. We provide examples for a class of utility functions.
期刊介绍：
SIAM Journal on Financial Mathematics (SIFIN) addresses theoretical developments in financial mathematics as well as breakthroughs in the computational challenges they encompass. The journal provides a common platform for scholars interested in the mathematical theory of finance as well as practitioners interested in rigorous treatments of the scientific computational issues related to implementation. On the theoretical side, the journal publishes articles with demonstrable mathematical developments motivated by models of modern finance. On the computational side, it publishes articles introducing new methods and algorithms representing significant (as opposed to incremental) improvements on the existing state of affairs of modern numerical implementations of applied financial mathematics.
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