On the homotopy analysis method for backward/forward-backward stochastic differential equations

TL;DR

This paper demonstrates that the homotopy analysis method (HAM) provides fast, accurate solutions for high-dimensional backward and forward-backward stochastic differential equations, outperforming existing numerical methods in efficiency.

Contribution

The paper introduces HAM as an effective analytic approximation technique for solving high-dimensional BSDEs and FBSDEs, showing improved computational efficiency and accuracy.

Findings

HAM achieves high accuracy solutions within 1% CPU time of existing methods.

HAM's computational complexity grows less rapidly with dimensionality.

Validated HAM's effectiveness in science, engineering, and finance applications.

Abstract

In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6 dimensional case, within less than $1\%$ CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering and finance.