In this paper, the objective is to study the continuous mean–variance portfolio selection with a no-short-selling constraint and obtain a time-consistent solution. We assume that there is a self-financing portfolio with wealth process [Formula: see text], in which [Formula: see text] represents the fraction of wealth invested in the risk asset under the short selling prohibition. We investigate the mean–variance optimal constrained problem defined by obtaining the supremum over all admissible controls of the difference between the expectation of the value process at some designated terminal time [Formula: see text] and a positive constant times the variance of [Formula: see text]. To envisage the quadratic nonlinearity introduced by the variance, the method of Lagrangian multipliers reduces the nonlinear problem into a set of linear problems which can be solved by applying the Hamilton–Jacobi–Bellman equation and change of variables formula with local time on curves. Solving the HJB system provides the time-inconsistent solution and from there, we derive the time-consistent optimal control.

中文翻译：

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无卖空约束的最优均值-方差投资组合选择

在本文中，目的是研究具有无卖空约束的连续均值-方差投资组合选择，并获得时间一致的解决方案。我们假设有一个具有财富过程的自筹资金组合[公式：见正文]，其中[公式：见正文]表示在卖空禁令下投资于风险资产的财富比例。我们研究通过在某个指定的终端时间[公式：见文本]的价值过程的期望与正常数乘以[公式：见正文]。为了设想方差引入的二次非线性，拉格朗日乘子法将非线性问题简化为一组线性问题，可以通过应用 Hamilton-Jacobi-Bellman 方程和曲线上的局部时间变量公式来求解。求解 HJB 系统提供了时间不一致的解决方案，并由此推导出时间一致的最优控制。