We examine nonlinear Kolmogorov partial differential equations (PDEs). Here the nonlinear part of the PDE comes from its Hamiltonian where one maximizes over all possible drift and diffusion coefficients which fall within a $\varepsilon$-neighborhood of pre-specified baseline coefficients. Our goal is to quantify and compute how sensitive those PDEs are to such a small nonlinearity, and then use the results to develop an efficient numerical method for their approximation. We show that as $\varepsilon\downarrow 0$, the nonlinear Kolmogorov PDE equals the linear Kolmogorov PDE defined with respect to the corresponding baseline coefficients plus $\varepsilon$ times a correction term which can be also characterized by the solution of another linear Kolmogorov PDE involving the baseline coefficients. As these linear Kolmogorov PDEs can be efficiently solved in high-dimensions by exploiting their Feynman-Kac representation, our derived sensitivity analysis then provides a Monte Carlo based numerical method which can efficiently solve these nonlinear Kolmogorov equations. We provide numerical examples in up to 100 dimensions to empirically demonstrate the applicability of our numerical method.

翻译：本文研究非线性Kolmogorov偏微分方程（PDE）。其非线性部分源于Hamiltonian函数，该函数对预设基线系数的$\varepsilon$-邻域内的所有可能漂移系数与扩散系数进行最大化。我们的目标是量化并计算此类方程对该小幅度非线性的敏感程度，进而利用分析结果建立高效的数值逼近方法。研究表明：当$\varepsilon\downarrow 0$时，非线性Kolmogorov PDE等于以相应基线系数定义的线性Kolmogorov PDE，再加上$\varepsilon$乘以一个可由另一含基线系数的线性Kolmogorov PDE解表征的修正项。利用Feynman-Kac表示可高效求解高维线性Kolmogorov PDE，因此本文推导的灵敏度分析提供了一种基于蒙特卡洛的数值方法，能够高效求解这些非线性Kolmogorov方程。我们提供了高达100维的数值算例，实证展示了所提数值方法的适用性。