Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method (Revised in January 2015; Forthcoming in "Asia-Pacific Financial Markets")

Abstract

In this paper, we propose an efficient Monte Carlo implementation of a non-linear FBSDE as a system of interacting particles inspired by the idea of the branching
diffusion method of McKean. It will be particularly useful to investigate large and
complex systems, and hence it is a good complement of our previous work presenting
an analytical perturbation procedure for generic non-linear FBSDEs. There appear multiple species of particles, where the first one follows the diffusion of the original
underlying state, and the others the Malliavin derivatives with a grading structure.
The number of branching points are capped by the order of perturbation, which is
expected to make the scheme less numerically intensive. The proposed method can be
applied to semi-linear problems, such as American Options, Credit and Funding Value Adjustments, and even fully non-linear issues, such as the optimal portfolio problems in incomplete and/or constrained markets.