{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:55:53Z","timestamp":1760057753000,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T00:00:00Z","timestamp":1739923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"General Project of the National Social Science Foundation of China, \u201cResearch on the Paths to Improve the Accessibility and Balance of Basic Public Services in Ethnic Minority Areas in Western China\u201d","award":["23BJL122","G202410595518"],"award-info":[{"award-number":["23BJL122","G202410595518"]}]},{"name":"Guilin University of Electronic Science and Technology Student Innovation and Entrepreneurship Project","award":["23BJL122","G202410595518"],"award-info":[{"award-number":["23BJL122","G202410595518"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox\u2013Ingersoll\u2013Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton\u2013Jacobi\u2013Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures.<\/jats:p>","DOI":"10.3390\/sym17020311","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T08:36:46Z","timestamp":1739954206000},"page":"311","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Optimal Investment Decision Problem Under the HARA Utility Framework"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4185-9914","authenticated-orcid":false,"given":"Aiyin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Statistical and Data Sciences, Xinjiang University of Finance and Economics, Urumqi 830012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Statistical and Data Sciences, Xinjiang University of Finance and Economics, Urumqi 830012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-2982","authenticated-orcid":false,"given":"Guodong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenjie","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,19]]},"reference":[{"key":"ref_1","first-page":"92","article-title":"Risks in Chinese household financial portfolios\u2014Too conservative or too aggressive?","volume":"12","author":"Lu","year":"2017","journal-title":"Manag. 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