We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.
{"title":"On the Expected Discounted Penalty Function Using Physics-Informed Neural Network","authors":"Jiayu Wang, Houchun Wang","doi":"10.1155/2023/9950023","DOIUrl":"https://doi.org/10.1155/2023/9950023","url":null,"abstract":"We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Muhammad Zainul Abidin, Muhammad Marwan, Naeem Ullah, Ahmed Mohamed Zidan
In this paper, we work on the Cauchy problem of the three-dimensional micropolar fluid equations. For small initial data, in the variable-exponent Fourier–Besov spaces, we achieve the global well-posedness result. The Littlewood–Paley decomposition method and the Fourier-localization technique are main tools to obtain the results. Moreover, the results discussed in our work show the Gevrey class regularity of solution to the Cauchy problem of micropolar fluid equations.
{"title":"Well-Posedness in Variable-Exponent Function Spaces for the Three-Dimensional Micropolar Fluid Equations","authors":"Muhammad Zainul Abidin, Muhammad Marwan, Naeem Ullah, Ahmed Mohamed Zidan","doi":"10.1155/2023/4083997","DOIUrl":"https://doi.org/10.1155/2023/4083997","url":null,"abstract":"In this paper, we work on the Cauchy problem of the three-dimensional micropolar fluid equations. For small initial data, in the variable-exponent Fourier–Besov spaces, we achieve the global well-posedness result. The Littlewood–Paley decomposition method and the Fourier-localization technique are main tools to obtain the results. Moreover, the results discussed in our work show the Gevrey class regularity of solution to the Cauchy problem of micropolar fluid equations.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a portfolio optimization problem where the wealth consists of investing into a risky asset with a slow mean-reverting volatility and receiving an uncontrollable stochastic cash flow under the exponential utility. The Hamilton–Jacobi–Bellman equation formulated from the optimal investment problem is a high-dimensional nonlinear partial differential equation and difficult to find its analytical or numerical solutions. The paper provides a tractable asymptotic approach which treats the initial problem as a perturbation around the constant volatility problem. In this paper, we present a formal derivation of asymptotic approximation and prove the accuracy of the value function. Moreover, an illustrative example is provided to assess our approximate strategy and value function.
{"title":"Application of Asymptotic Analysis of a High-Dimensional HJB Equation to Portfolio Optimization","authors":"Lei Hu","doi":"10.1155/2023/3399493","DOIUrl":"https://doi.org/10.1155/2023/3399493","url":null,"abstract":"In this paper, we consider a portfolio optimization problem where the wealth consists of investing into a risky asset with a slow mean-reverting volatility and receiving an uncontrollable stochastic cash flow under the exponential utility. The Hamilton–Jacobi–Bellman equation formulated from the optimal investment problem is a high-dimensional nonlinear partial differential equation and difficult to find its analytical or numerical solutions. The paper provides a tractable asymptotic approach which treats the initial problem as a perturbation around the constant volatility problem. In this paper, we present a formal derivation of asymptotic approximation and prove the accuracy of the value function. Moreover, an illustrative example is provided to assess our approximate strategy and value function.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138824741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Research on the Spread Path and Evolution Causes of Oral Language in the Digital Era","authors":"Journal of Mathematics","doi":"10.1155/2023/9878209","DOIUrl":"https://doi.org/10.1155/2023/9878209","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"123 35","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138953713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Classification and Analysis of College Students’ Skills Using Hybrid AI Models","authors":"Journal of Mathematics","doi":"10.1155/2023/9842979","DOIUrl":"https://doi.org/10.1155/2023/9842979","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"62 21","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138957063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Optimization Algorithm of Logistics Transportation Cost of Prefabricated Building Components for Project Management","authors":"Journal of Mathematics","doi":"10.1155/2023/9858351","DOIUrl":"https://doi.org/10.1155/2023/9858351","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"24 11","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138955168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Design of Enterprise Economic Information Management System Based on Big Data Integration Algorithm","authors":"Journal of Mathematics","doi":"10.1155/2023/9876921","DOIUrl":"https://doi.org/10.1155/2023/9876921","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"18 2","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138994351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Text Knowledge Acquisition Method of Collaborative Product Design Based on Genetic Algorithm","authors":"Journal of Mathematics","doi":"10.1155/2023/9894560","DOIUrl":"https://doi.org/10.1155/2023/9894560","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"32 16","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138955894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Evaluation Algorithm of Labor Legal Effectiveness for Affirmative Action against Gender Discrimination","authors":"Journal of Mathematics","doi":"10.1155/2023/9892024","DOIUrl":"https://doi.org/10.1155/2023/9892024","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"56 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138957109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Data Mining and Analysis of the Compatibility Law of Traditional Chinese Medicines Based on FP-Growth Algorithm","authors":"Journal of Mathematics","doi":"10.1155/2023/9761056","DOIUrl":"https://doi.org/10.1155/2023/9761056","url":null,"abstract":"<jats:p />","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"41 24","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138957208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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