Functional Itô calculus
将伊藤积分推广到依赖于路径的泛函，定义了新的偏导数并推导出伊藤公式，还扩展了Feynman-Kac公式，发现路径依赖期权价格局部满足偏微分方程，可用于计算波动率曲面的变分导数。
We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.