考虑一般约束条件下的消费投资决策模型研究

This research project contributes to the theory of optimal consumption-investment in intertemporal economies. The optimal consumption-investment problem studies the decisions of an agent endowed with some initial wealth who seeks to maximize the expected utility of consumption. These decisions are the consumption rate and the allocation of his wealth to risky and risk-free assets (known as investment strategy) over time. The classical papers of Samuelson (1969) and Merton (1969, 1971) began the study of dynamic optimal consumption-investment problems. Models involving general constraints have since been investigated. In this project, we plan to study the optimal consumption-investment problem, in which the investment strategy is constrained and the consumption rate is subject to an upper and/or lower boundary at any time. The boundaries are assumed to be wealth dependent. An example of an investor that resembles the aforementioned agent is an investment firm with cash flow commitments that is subject to regulatory constraints. We will consider the problem in the context of the standard Black-Scholes-Merton market. The goal of this project is to determine the value function of the problem, to examine how smooth it is, and to determine the optimal investment strategy. The main tools are coming from the theory of differential equations, particularly the theories of free boundary and viscosity solution for differential equations. We will first demonstrate that the value function is the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We will then prove that the viscosity solution of the equation is smooth. Finally, the optimal consumption rate and investment strategy are provided in feedback form of the wealth. The international financial crises that occurred in the past few years has forced governments to impose a series of strict regulations on investment and trading to eliminate or mitigate financial crises. Although extensive research on the optimal consumption-investment problem has been conducted, research on this problem under the constraint that the consumption rate is subject to an upper and/or lower boundary at any time is scant. Consequently, this research topic has not been sufficiently explored. By establishing and analyzing new models that can be used in this field, we expect that the findings of this research project will help investors and financial institutions obtain a more comprehensive understanding of financial investment and risk control.

本项目考虑连续时间下最优投资消费模型。此问题研究投资者面临选择最优的投资消费策略以最大化其期望消费效用。决策变量为消费策略以及投资策略。Samuelson 和Merton开创了动态最优投资消费问题的研究。此后大量文献对此问题进行了深入的研究。本项目将采用 Black-Scholes模型假设，对消费率依赖于财富上下界限制的投资消费问题进行研究。常见的例子包括基于监管要求投资者只能把部分财富投资到风险资产上面的情形。项目目标是决定值函数并给出最优投资消费策略。主要工具来自随机控制以及微分方程领域，特别是黏性解及自由边界理论。我们希望证明值函数是相应的 HJB方程唯一的光滑黏性解，最后得到一个反馈形式的最优投资消费策略。近期发生的金融危机使得国际社会加强了对投资消费的监管措施，以减小甚至避免金融危机重来。我们希望本项目的研究成果能够帮助投资者进行更合理的投资消费，并给监管部门提供一定参考意见。

经典的期望方差理论及期望效用理论是现代金融学的两大基石。与之相关的投资组合问题一直是金融领域研究的重点、核心。本项目主要是研究带有约束条件的最优消费投资问题，以及与之相关的偏微分方程及随机控制问题。项目研究了带有下约束、上约束、以及综合约束条件的消费投资模型，同时考虑了更为广泛的类似问题。通过对此类问题的研究，对已有的自由边界理论和随机控制理论进行了进一步的改进和发展。本项目按照原计划执行了研究工作，完成了既定目标，已经发表及接受了十五篇文章，获得了一些重要成果。依托本项目，培养了两位博士研究生。