  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.
