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A novel second order scheme with one step for forward backward stochastic differential equations

Qiang Han, Shihao Lan, Quanxin Zhu

TL;DR

The paper tackles numerical solution of decoupled forward-backward stochastic differential equations (FBSDEs) by introducing a novel explicit one-step second-order discretization parameterized by $α ∈ (0,1]$, which recovers Crank–Nicolson when $α=1$. It provides a stability analysis and local truncation error bounds via Itô–Taylor expansions, leading to a global error bound that confirms second-order convergence. Numerical experiments on a scalar BSDE and a FitzHugh–Nagumo–type FBSDE validate the theory, showing near–two order convergence across different $α$ values and demonstrating that the method remains efficient and fully explicit for both $Y$ and $Z$. The work offers a practical, high-order alternative to multistep or implicit schemes and lays groundwork for constructing higher-order one-step methods.

Abstract

In this paper, we present a novel explicit second order scheme with one step for solving the forward backward stochastic differential equations, with the Crank-Nicolson method as a specific instance within our proposed framework. We first present a rigorous stability result, followed by precise error estimates that confirm the proposed novel scheme achieves second-order convergence. The theoretical results for the proposed methods are supported by numerical experiments.

A novel second order scheme with one step for forward backward stochastic differential equations

TL;DR

The paper tackles numerical solution of decoupled forward-backward stochastic differential equations (FBSDEs) by introducing a novel explicit one-step second-order discretization parameterized by , which recovers Crank–Nicolson when . It provides a stability analysis and local truncation error bounds via Itô–Taylor expansions, leading to a global error bound that confirms second-order convergence. Numerical experiments on a scalar BSDE and a FitzHugh–Nagumo–type FBSDE validate the theory, showing near–two order convergence across different values and demonstrating that the method remains efficient and fully explicit for both and . The work offers a practical, high-order alternative to multistep or implicit schemes and lays groundwork for constructing higher-order one-step methods.

Abstract

In this paper, we present a novel explicit second order scheme with one step for solving the forward backward stochastic differential equations, with the Crank-Nicolson method as a specific instance within our proposed framework. We first present a rigorous stability result, followed by precise error estimates that confirm the proposed novel scheme achieves second-order convergence. The theoretical results for the proposed methods are supported by numerical experiments.
Paper Structure (6 sections, 6 theorems, 89 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 6 sections, 6 theorems, 89 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Novel one step discretization of the BSDE
  3. Stability analysis
  4. Error estimates
  5. Numerical experiments
  6. Conclusions

Key Result

Theorem 3.1

Suppose the assumptions (i)-(iii) and $f(t,Y_t,Z_t)\in C_b^3$ hold. Then for a sufficiently small step size $h_t$, we have where $C$ denotes a positive constant.

Figures (3)

  • Figure 1: The plot of $\text{err}_{y}$ against time interval $\Delta t$ in Example 1.
  • Figure 2: Log-log plot of $\text{err}_{y}$ against time interval $\Delta t$ in Example 2 ($a=-0.5$, and $x_0=1$).
  • Figure 3: Log-log plot of $\text{err}_{y}$ against time interval $\Delta t$ in Example 2 ($a=-1$, and $x_0=1.5$).

Theorems & Definitions (6)

  • Theorem 3.1
  • Proposition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5