非期望效用理论框架下的金融随机优化问题研究

Behavioral finance is a hot research topic in modern finance, and the main achievements,the rank dependent utility theory and prospect theory,attract a great deal of attention. Under the framework of these non-expected utility theories, how to construct investment decision models realistically, how to derive the specific quantization scheme, and how to instruct people to make reasonable investment decisions, have brought new challenges to stochastic optimization theory. Under the framework of non-expected utility theory, this project studies stock and option portfolio selection, and option pricing, and the corresponding stochastic optimiziton problem by using quantile function formulation, Wang-transform and other techniques dealing with the probability distortion problems and the optimization problems for non-convex/non-concave functions, combined with stochastic control theory, nonlinear expectation theory, risk measure theory and mathematical finance theory. The main issues investigated are as follows: optimal strategies for stock investment based on non-expected theory, optimal investment strategies for holding option positions, optimal position for options, and the model of option pricing based on the portfolio selection models by using the indifference pricing principle and factorization of risk.The research of the project not only enriches the mathematical finance theory, but also promotes the development of stochastic optimization theory.

行为金融是现代金融理论的研究热点，作为主要成果的秩相依效用理论和前景理论更是引起广泛的关注，如何在这些非期望效用理论框架下建立符合实际的投资决策模型，给出具体的量化方案，指导人们的投资决策，给随机优化理论带来了新的挑战。本项目将在非期望效用理论框架下，利用分位数函数、王-转换等处理概率扭曲问题的技术以及处理非凸/非凹函数的最优化问题的方法，结合随机控制、非线性期望、风险度量和数理金融方面的理论成果，研究金融市场中的股票、期权投资组合选择和期权定价问题，系统探讨这些问题中涉及到的随机优化难题。主要内容包括：基于非期望效用理论的最优股票投资策略；考虑持有期权头寸的投资策略；期权的最优持有头寸以及利用无差异定价原理、风险分解的思想等等，在投资组合模型基础上建立非期望效用理论下的期权定价模型。本项目的研究不仅可以充实现有的数理金融理论，也将推动随机优化理论的发展。

现代金融保险中的随机优化问题是近年来在数理金融领域研究的前沿热点问题。本项目致力于在非期望效用理论框架下，建立符合实际的投资决策模型，运用随机分析的相关理论和方法，研究投资者的最优投资及相关的随机优化问题。我们的研究取得如下研究成果。首先，讨论了一类跳扩散金融市场下的损失厌恶投资者的最优投资策略，利用鞅方法和最优化技巧，解决了其中的随机优化问题；其次，研究了秩相依效用理论框架下的最优投资和再保险问题；另外，我们研究了保险公司投资于常弹性方差金融市场和购买比例-超额损失组合再保险的最优投资策略；以及以破产概率最小化为目标，考虑公司面临两类保险业务下的最优再保险问题；还有，我们讨论了指数均值回复金融市场下的最优投资策略和最优再保险问题。研究结果充实了现有的数理金融理论，也为金融保险行业的决策提供了有价值的依据。