在国民经济发展中，尤其是在金融产品创新、金融风险管理等领域，有很多金融问题亟待解决。而解决这些现代的金融问题，离不开定量分析，这就需要引入、创新相关的数学理论与方法。本项目就是在这一问题背景下，拟利用和发展随机微分方程、随机最优控制、偏微分方程和数值分析等数学理论和方法，研究随机和不确定波动率下的金融衍生品尤其是若干奇异期权的定价。我们尝试探讨不确定波动率下非标准期权的定价方程及求解，波动率具有时滞情形下的最优投资问题，多因子Gauss随机波动率模型下的目标波动率期权定价以及摆动期权的定价求解问题。这些问题若能得到解决，不仅会促进金融衍生产品创新，加强人们对金融衍生产品的认知和风险管理，也会发展和引入新的数学理论与方法，并加深人们对随机和不确定性这一客观现象的认知。

In the development of national economy, especially in the areas of financial products innovation and financial risk management, there are still numerous financial problems need to be solved urgently. In order to solve these modern financial problems, quantitative analysis is indispensable, which requires introduction and innovation of the relevant mathematical theories and methods. Under the background of these problems, this project is proposed to introduce and develop the theories and methods of stochastic differential equations, stochastic optimal control, partial differential equations and numerical analysis to research the pricing principle of financial derivatives with the stochastic and uncertain volatilities, especially some exotic options pricing. In this project, we try to explore the non-standard option pricing equations and solutions under the uncertainty volatility, the optimal portfolio problems when volatility has the time delay, the pricing and the corresponding closed form solutions of the target volatility options and swing options under the multi-factor Gauss stochastic volatility model. If these problems can be solved, not only the innovation of financial derivatives will be promoted to strengthen people's perception of financial derivatives and risk management, but also some new mathematical methods will be developed and introduced, and deepening people's understanding of the essential of random and uncertainty.