High-order combined Multi-step Scheme for solving forward Backward Stochastic Differential Equations

TL;DR

This paper introduces a high-order multi-step scheme for solving forward-backward stochastic differential equations, achieving up to ninth-order convergence and enhanced efficiency through advanced numerical techniques.

Contribution

The paper develops a novel high-order multi-step method combining finite differences, Gaussian quadrature, and polynomial interpolation for FBSDEs, improving convergence rates.

Findings

Achieves convergence rate up to ninth order.

Demonstrates improved efficiency over existing methods.

Provides numerical validation of convergence.

Abstract

In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we adopt the high-order multi-step method in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36(4) (2014), pp.A1731-A1751] by combining multi-steps. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the time direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.