Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme (Revised as CARF-F-269)

Abstract

In this work, we have presented a simple analytical approximation scheme for the generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for the standard European contingent claims. 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 the perturbation works and show its accuracy relative to the known numerical techniques. The method presented in this paper may be useful for various important problems which have prevented analytical treatment so far.