均值-方差投资组合选择：初终值成本耦合下均值场系统与正倒向随机系统的联系
Mean-variance portfolio selection: connections between mean-field systems and forward-backward stochastic systems under initial-terminal cost coupling
- Hong Kong Polytechnic University（香港理工大学）
- Shandong University（山东大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过引入终端财富条件期望，将均值-方差投资组合选择问题转化为初终值耦合的正倒向随机系统控制问题，建立随机最大值原理并推导广义LQ控制的最优反馈，显式给出有效策略与前沿。
AI中文摘要：
本文研究了由均值-方差投资组合选择问题产生的均值场系统与正倒向随机系统之间的联系。通过引入终端财富的条件期望过程，均值场项被表示为后向状态，从而引出一类具有初终值耦合的正倒向随机系统控制问题。成本泛函中前向状态终值与后向状态初值之间的耦合使得经典的正倒向随机最优控制方法无法直接应用。为克服这一困难，我们针对具有运行成本和耦合初终值的控制问题建立了随机最大值原理，并给出了最优性的充分条件。此外，我们研究了一类具有初终值耦合的广义随机线性二次控制问题，允许运行成本中的控制权重为不定矩阵，并通过Riccati方程推导了最优反馈表示。经典均值-方差投资组合选择模型被证明是特例，我们显式地得到了相应的有效投资策略和有效前沿，并通过数值模拟验证了结果，检验了参数敏感性及其金融含义。
英文摘要：
This paper investigates the connection between mean-field systems and forward-backward stochastic systems arising from the mean-variance portfolio selection problem. By introducing the conditional expectation process of terminal wealth, the mean-field term is represented by a backward state, leading to a class of control problems for forward-backward stochastic systems with coupled initial-terminal values. The coupling between the terminal value of the forward state and the initial value of the backward state in the cost functional prevents classical methods for forward-backward stochastic optimal control from being directly applicable. To overcome this difficulty, we establish a stochastic maximum principle for control problems with running costs and coupled initial-terminal values, and provide sufficient conditions for optimality. Furthermore, we investigate a class of generalized stochastic linear-quadratic control problems with initial-terminal coupling, allowing the control weight in the running cost to be indefinite, and derive the optimal feedback representation via a Riccati equation. The classical mean-variance portfolio selection model is shown to be a special case, for which the corresponding efficient investment strategy and efficient frontier are obtained explicitly, and numerical simulations are presented to illustrate the results and examine parameter sensitivity and its financial implications.
