Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method

TL;DR

This paper introduces a Monte Carlo particle-based method for solving complex non-linear FBSDEs, leveraging perturbation techniques to improve computational efficiency and applicability to various financial problems.

Contribution

It presents a novel interacting particle method for non-linear FBSDEs that reduces computational complexity using perturbation and branching techniques.

Findings

Efficient Monte Carlo implementation for non-linear FBSDEs.

Applicable to semi-linear and fully non-linear financial problems.

Reduces numerical intensity via perturbation order capping.

Abstract

In this paper, we propose an efficient Monte Carlo implementation of non-linear FBSDEs as a system of interacting particles inspired by the ideas of branching diffusion method. 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 and Bermudan options, Credit Value Adjustment (CVA), and even fully non-linear issues, such as the optimal portfolio problems in incomplete and/or constrained markets, feedbacks from large investors, and also the analysis of various risk measures.