{"title":"时间不一致控制的弱均衡:在投资退出决策中的应用","authors":"Zongxia Liang, Fengyi Yuan","doi":"10.1111/mafi.12391","DOIUrl":null,"url":null,"abstract":"<p>This paper considers time-inconsistent problems when control and stopping strategies are required to be made <i>simultaneously</i> (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mover>\n <mi>u</mi>\n <mo>̂</mo>\n </mover>\n <mo>,</mo>\n <mi>C</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\hat{u},C)$</annotation>\n </semantics></math> of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to <i>verify</i> or <i>exclude</i> equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":1.6000,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions\",\"authors\":\"Zongxia Liang, Fengyi Yuan\",\"doi\":\"10.1111/mafi.12391\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper considers time-inconsistent problems when control and stopping strategies are required to be made <i>simultaneously</i> (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mover>\\n <mi>u</mi>\\n <mo>̂</mo>\\n </mover>\\n <mo>,</mo>\\n <mi>C</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\hat{u},C)$</annotation>\\n </semantics></math> of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to <i>verify</i> or <i>exclude</i> equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.</p>\",\"PeriodicalId\":49867,\"journal\":{\"name\":\"Mathematical Finance\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.6000,\"publicationDate\":\"2023-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Finance\",\"FirstCategoryId\":\"96\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/mafi.12391\",\"RegionNum\":3,\"RegionCategory\":\"经济学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"BUSINESS, FINANCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Finance","FirstCategoryId":"96","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/mafi.12391","RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"BUSINESS, FINANCE","Score":null,"Total":0}
Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions
This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.
期刊介绍:
Mathematical Finance seeks to publish original research articles focused on the development and application of novel mathematical and statistical methods for the analysis of financial problems.
The journal welcomes contributions on new statistical methods for the analysis of financial problems. Empirical results will be appropriate to the extent that they illustrate a statistical technique, validate a model or provide insight into a financial problem. Papers whose main contribution rests on empirical results derived with standard approaches will not be considered.