Rough随机波动率模型的金融应用及算法研究

本项目拟研究Rough随机波动率模型在金融市场中的应用及算法研究。主要研究：使用时间多尺度Hawkes过程推导Rough随机波动率模型，设计Rough随机波动率模型快速模拟算法， 推导Rough随机波动率模型期权定价的时间分数阶偏微分方程，研究该时间分数阶偏微分方程的数值方法和收敛阶理论分析，研究该模型期权定价的连续时间马尔科夫链逼近算法，建立Rough随机波动率模型期权定价计算和实证的求解器或程序包，研究Rough随机波动率模型最优投资（期望效用最大化问题）的快速算法，推导该模型最优投资问题的时间分数阶HJB方程，并研究该时间分数阶HJB方程的有限差分方法和收敛理论分析。

The aim of this project is to study the financial applications of the rough stochastic volatility models and the algorithms therein. The main contents include: The variants of the rough stochastic volatility models will be derived using the multiscale-time Hawkes processes. The fast algorithms for simulating the random paths of the rough stochastic volatility models will be designed. The time fractional partial differential equations for the option pricing under the rough stochastic volatility models will be derived and solved by the numerical methods and the convergence rates will be proved. Also the continuous-time Markov chain approximation will be studied for the option pricing under the models. The solvers or codes packages will be built for the option pricing and empirical analysis of the models. The fast algorithms for the optimal investment (expectation utility maximization) under the rough stochastic volatility models will be studied. The time fractional HJB equations for the optimal investment will be derived, the finite difference methods will be proposed to solve the equations and the convergence theory will be analyzed.