Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme

TL;DR

This paper introduces a recursive analytical approximation scheme for non-linear FBSDEs by perturbing around linear decoupled systems, enabling solutions via standard PDE methods and demonstrating accuracy through examples.

Contribution

It presents a novel perturbation-based analytical approximation method for non-linear FBSDEs and PDEs, bridging probabilistic and PDE approaches for easier solutions.

Findings

The method converts non-linear PDEs into linear parabolic PDEs.

Recursive approximation achieves arbitrarily high order accuracy.

Examples show the method's effectiveness compared to numerical techniques.

Abstract

In this work, we have presented a simple analytical approximation scheme for 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 a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is 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 are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far.