本项目探究行为金融中连续时间的最优资产组合问题，特别是结合累积展望理论且包含形式多样的金融活动的最优控制问题。累积展望理论把认知心理结合到风险决策中，给传统的投资数学模型带来巨大挑战。对此，本项目一方面结合累积展望理论研究具体的投资/消费决策问题，在随机最大值原理的基础上，处理效用函数因S形和零点导数无穷大带来的不良性质，以及应对概率扭曲产生的随机变量P(X≤Y)（X,Y同分布），得出最优行为策略。然后利用其显式解，分析人的风险喜恶与概率扭曲程度，如何影响最优策略选择。另一方面，逐步发展结合累积展望理论的控制系统，先联合Markov机制转换和跳跃过程，再讨论更一般化且含有参照点的行为决策模型。这些模型的最优化准则将被应用到具体例子中，以提供投资人最优行为策略，进一步地，借此揭示人在金融活动中的一些行为规律。

This project is to study the continuous-time optimal portfolio problems in behavioral finance, in particular, the optimal control problems involved cumulative prospect theory (CPT) as well as various financial activities. CPT incorporates cognitive psychology into risk decision problems, which brings in huge challenges to conventional investment mathematical models. To this end, the project considers an investment/consumption selection problem combined with CPT on one hand. In addition to stochastic maximum principle, the project deals with ill-behaved utility function due to its S-shape and infinite derivative at 0 and handles of random variable P(X≤Y)（X,Y share the same distribution）on account of probability distortion, so that an optimal behavioral portfolio can be derived. With the explicit solution, it will analyze how changes in risk aversion/seeking and probability distortion influence optimal portfolio selection. On the other hand, the project will develop the CPT control system step by step. It will be combined with Markov regime-switching and jump process at first. Then a more general behavioral decision model containing reference point will be discussed. The optimality principles of these models would be applied to specific cases, so as to provide optimal behavioral strategies for investors. In this way, the project tries to further reveal some rules of human behavior in financial activities.