New Second-order Convergent Schemes for Solving decoupled FBSDEs

TL;DR

This work tackles the numerical solution of decoupled forward-backward stochastic differential equations by introducing two second-order symmetric ADI-type schemes that split the generator $f$ into $f_1+f_2$. The value process $Y$ is evolved using alternating explicit/implicit updates on the split components, while the $Z$-process leverages the ZhaoLi2014 schemes. The authors prove second-order convergence and demonstrate computational savings when the generator comprises a linear part plus a nonlinear part, supported by numerical tests including a backward stochastic Riccati equation from mean-variance hedging. The results offer a practical approach to accelerate BSDE solvers without sacrificing accuracy, with potential impact on finance and stochastic control applications.

Abstract

This paper proposes a new second-order symmetric algorithm for solving decoupled forward-backward stochastic differential equations. Inspired by the alternating direction implicit splitting method for partial differential equations, we split the generator into the sum of two functions. In the computation of the value process Y, explicit and implicit schemes are alternately applied to these two generators, while the algorithms from \citep{ZhaoLi2014} are used for the control process Z. We rigorously prove that the two new schemes have second-order convergence rate. The proposed splitting methods show clear advantages for equations whose generator consists of a linear part plus a nonlinear part, as they reduce the number of iterations required for solving implicit schemes, thereby decreasing computational cost while maintaining second-order convergence. Two numerical examples are provided, including the backward stochastic Riccati equation arising in mean-variance hedging. The numerical results verify the theoretical error analysis and demonstrate the advantage of reduced computational cost compared to the algorithm in \citep{ZhaoLi2014}.

Theorems & Definitions (11)

• Lemma 2.1: See nualart1997
• Lemma 2.2: See nualart1997
• Theorem 3.1
• proof
• Theorem 3.2
• Lemma 3.1
• Lemma 3.2
• proof
• proof : Proof of \ref{['Theorem3.2']}
• Example 4.1