{"title":"最优消费——时变不完全偏好下的投资问题","authors":"Weixuan Xia","doi":"arxiv-2312.00266","DOIUrl":null,"url":null,"abstract":"The main objective of this paper is to develop a martingale-type solution to\noptimal consumption--investment choice problems ([Merton, 1969] and [Merton,\n1971]) under time-varying incomplete preferences driven by externalities such\nas patience, socialization effects, and market volatility. The market is\ncomposed of multiple risky assets and multiple consumption goods, while in\naddition there are multiple fluctuating preference parameters with inexact\nvalues connected to imprecise tastes. Utility maximization is a multi-criteria\nproblem with possibly function-valued criteria. To come up with a complete\ncharacterization of the solutions, first we motivate and introduce a set-valued\nstochastic process for the dynamics of multi-utility indices and formulate the\noptimization problem in a topological vector space. Then, we modify a classical\nscalarization method allowing for infiniteness and randomness in dimensions and\nprove results of equivalence to the original problem. Illustrative examples are\ngiven to demonstrate practical interests and method applicability\nprogressively. The link between the original problem and a dual problem is also\ndiscussed, relatively briefly. Finally, using Malliavin calculus with\nstochastic geometry, we find optimal investment policies to be generally\nset-valued, each of whose selectors admits a four-way decomposition involving\nan additional indecisiveness risk-hedging portfolio. Our results touch on new\ndirections for optimal consumption--investment choices in the presence of\nincomparability and time inconsistency, also signaling potentially testable\nassumptions on the variability of asset prices. Simulation techniques for\nset-valued processes are studied for how solved optimal policies can be\ncomputed in practice.","PeriodicalId":501487,"journal":{"name":"arXiv - QuantFin - Economics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences\",\"authors\":\"Weixuan Xia\",\"doi\":\"arxiv-2312.00266\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main objective of this paper is to develop a martingale-type solution to\\noptimal consumption--investment choice problems ([Merton, 1969] and [Merton,\\n1971]) under time-varying incomplete preferences driven by externalities such\\nas patience, socialization effects, and market volatility. The market is\\ncomposed of multiple risky assets and multiple consumption goods, while in\\naddition there are multiple fluctuating preference parameters with inexact\\nvalues connected to imprecise tastes. Utility maximization is a multi-criteria\\nproblem with possibly function-valued criteria. To come up with a complete\\ncharacterization of the solutions, first we motivate and introduce a set-valued\\nstochastic process for the dynamics of multi-utility indices and formulate the\\noptimization problem in a topological vector space. Then, we modify a classical\\nscalarization method allowing for infiniteness and randomness in dimensions and\\nprove results of equivalence to the original problem. Illustrative examples are\\ngiven to demonstrate practical interests and method applicability\\nprogressively. The link between the original problem and a dual problem is also\\ndiscussed, relatively briefly. Finally, using Malliavin calculus with\\nstochastic geometry, we find optimal investment policies to be generally\\nset-valued, each of whose selectors admits a four-way decomposition involving\\nan additional indecisiveness risk-hedging portfolio. Our results touch on new\\ndirections for optimal consumption--investment choices in the presence of\\nincomparability and time inconsistency, also signaling potentially testable\\nassumptions on the variability of asset prices. Simulation techniques for\\nset-valued processes are studied for how solved optimal policies can be\\ncomputed in practice.\",\"PeriodicalId\":501487,\"journal\":{\"name\":\"arXiv - QuantFin - Economics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - QuantFin - Economics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2312.00266\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuantFin - Economics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2312.00266","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences
The main objective of this paper is to develop a martingale-type solution to
optimal consumption--investment choice problems ([Merton, 1969] and [Merton,
1971]) under time-varying incomplete preferences driven by externalities such
as patience, socialization effects, and market volatility. The market is
composed of multiple risky assets and multiple consumption goods, while in
addition there are multiple fluctuating preference parameters with inexact
values connected to imprecise tastes. Utility maximization is a multi-criteria
problem with possibly function-valued criteria. To come up with a complete
characterization of the solutions, first we motivate and introduce a set-valued
stochastic process for the dynamics of multi-utility indices and formulate the
optimization problem in a topological vector space. Then, we modify a classical
scalarization method allowing for infiniteness and randomness in dimensions and
prove results of equivalence to the original problem. Illustrative examples are
given to demonstrate practical interests and method applicability
progressively. The link between the original problem and a dual problem is also
discussed, relatively briefly. Finally, using Malliavin calculus with
stochastic geometry, we find optimal investment policies to be generally
set-valued, each of whose selectors admits a four-way decomposition involving
an additional indecisiveness risk-hedging portfolio. Our results touch on new
directions for optimal consumption--investment choices in the presence of
incomparability and time inconsistency, also signaling potentially testable
assumptions on the variability of asset prices. Simulation techniques for
set-valued processes are studied for how solved optimal policies can be
computed in practice.