Before revision:  We have also provided two concrete examples to demonstrate how it works and shown its accuracy relative to the results directly obtained from numerical techniques, such as PDE and Monte Carlo simulation.
After revision:  We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method was found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples were provided to demonstrate how it works and shown its accuracy relative to the results directly obtained from numerical techniques, such as PDE and Monte Carlo simulation.