存在交易费用的非凹期望效用理论与资产最优化问题

Expected utility theory is one of the core contents of modern financial mathematics. It mainly studies how investors allocate assets to maximize the expected utility on their terminal wealth. It not only provides a theoretic basis for investors to make reasonable decisions and avoid risks, but also provides methods to set up pricing mechanism for financial assets. This project is to extend the traditional expected utility theory based on the assumption of investors' risk aversion and consider utility maximization problems with non-concave utility functions. Simultaneously, with the consideration of transaction costs in the real market, the project seeks optimal trading strategies under a reasonable given market structure. The following specific contents are included in this project: (1) Establish a continuous-time model, and prove the existence of solutions to the maximization problem of investor's expected utility in the framework that non-concave utility functions are defined on the positive real axis. Furthermore, utility-based pricing is derived. (2) Establish a discrete-time model, then prove the solvability of the corresponding problems by applying the so-called dynamic programming principle. This project generalizes the traditional expected utility theory, breaks through the hypothesis that investors are risk aversion, establishes models closer to the real market, and improves the description of investors' behavior which is more in line with that in the prospect theory. The expected research results of this project is to provide a theoretical basis for further research on investor behavior and asset allocation assuming that investors have variable preferences.

期望效用理论是现代金融的核心内容之一，主要研究投资人如何通过配置资产来实现期望效用最大化，它为投资人的投资决策和风险规避提供了科学的依据，并为资产定价提供了新颖的角度。本项目将拓展传统的以风险厌恶型偏好为基础的期望效用理论，考虑效用被非凹函数刻画的一类最大化问题。同时将考虑真实市场存在交易费用的因素，在给定合理的市场结构性条件下，寻求这类最大化问题的最优解。本项目的具体内容包括：（一）建立连续时间模型，在不允许做空的框架下构建极大化序列证明投资人期望效用最大化问题的解的存在性，由此可进行效用无差别定价；(二）建立离散时间模型，应用动态规划原理证明该问题的可解性。本项目的研究一般化了传统的效用理论，突破了投资人满足风险厌恶的假设前提，提出的模型更加贴近真实市场，对投资行为的描述切合前景理论。本项目的研究成果将为进一步研究以可变效用为背景的投资者行为和资产配置提供理论基础。

期望效用理论主要研究投资人如何通过配置资产来实现期望效用最大化。本项目以更广泛的风险偏好为基础研究了存在交易费用以及价值清算情况下的期望效用最大化问题，尤其是最优投资策略的存在性。从传统的期望效用最大化的结论出发，我们成功的通过研究效用最大化问题的稳定性得到了极大化序列的构造方式、收敛条件和收敛方式，并以此为基础，以效用函数的凸共轭为桥梁构造了使得非凹情况下效用最大化问题有最优策略的结构性条件，搭建了金融模型，证明了存在一个最优策略使其期望效用达到最大。之后还研究了价值函数的性质以及效用函数和它的凹包络各自的价值函数之间的关系，发现它的凹包络的价值函数等同于效用函数本身的价值函数的凹包络。在研究凹包络下的效用最大化问题时，我们纠正了在投资风险资产情况下的效用最大化问题的一些结论。本项目的研究成果为进一步研究以可变效用为背景的投资者行为和资产配置提供理论基础和方法。最后把项目的优化理念推广到民生领域，结合变点和孤立森林算法用于企业电力的异常检测，使得能更有效的监管企业的安全用电。