带跳随机微分方程的灵敏度分析及应用

Behavioural decisions in financial risk management and stochastic controls demand sensitivity analysis for stochastic differential equations with jumps. The project is to conduct sensitivity analysis and establish Greeks formulae for stochastic differential equations driven by multiplicative Lévy noises including α-stable-like noises by using Bismut's approach to Malliavin calculus with jumps. Numerical approximation and theoretical error analysis for Greeks are carried out by means of Euler-Maruyama method or other improved algorithm. As an application of sensitivity analysis for stochastic differential equations with jumps, numerical simulation for Greeks of options pricing is performed with the method basing on Malliavin calculus with jumps and Monte Carlo finite difference approach. In addition, we study stochastic optimal control utilizing stochastic gradient algorithm.

金融风险管理与随机控制中的行为决策需要考虑带跳随机微分方程的灵敏度分析。为此，本项目拟借助于带跳Malliavin计算中的Bismut方法，对一类由可乘Lévy过程（包括α稳定过程）驱动的含参随机微分方程进行灵敏度分析，建立相应的Greeks公式。在此基础上，利用Euler-Maruyama方法或其他改进的可行算法，对所建立的Greeks公式进行数值逼近和理论误差分析。作为带跳随机微分方程灵敏度分析的应用，我们拟利用基于带跳Malliavin计算的方法和蒙特卡罗有限差分方法对期权定价问题中的Greeks进行数值模拟。此外，我们还拟利用随机梯度算法研究随机最优控制问题。