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Convergence of the Markovian iteration for coupled FBSDEs via a differentiation approach

Zhipeng Huang, Cornelis W. Oosterlee

TL;DR

The paper tackles solving fully Z-coupled FBSDEs by extending the Markovian iteration with a differentiation-based treatment of the Z-term, enabling control of decoupling-field Lipschitz constants and well-posedness of the discretized scheme. The authors establish uniform Lipschitz and linear-growth bounds for the decoupling fields, derive time-discretization error estimates, and prove convergence of the Markovian iterations under generalized weak/monotone conditions. They also show that enforcing the relation $v= abla_x u\cdot\sigma$ within a single backward regression is crucial for stability, and numerical experiments confirm convergence and accuracy, highlighting advantages over direct extension methods. The results provide a rigorous foundation for solving Z-coupled FBSDEs and support the applicability of related Deep BSDE approaches in this more general setting.

Abstract

In this paper, we investigate the Markovian iteration method for solving coupled forward-backward stochastic differential equations (FBSDEs) featuring a fully coupled forward drift, meaning the drift term explicitly depends on both the forward and backward processes. An FBSDE system typically involves three stochastic processes: the forward process $X$, the backward process $Y$ representing the solution, and the $Z$ process corresponding to the scaled derivative of $Y$. Prior research by Bender and Zhang (2008) has established convergence results for iterative schemes dealing with $Y$-coupled FBSDEs. However, extending these results to equations with $Z$ coupling poses significant challenges, especially in uniformly controlling the Lipschitz constant of the decoupling fields across iterations and time steps within a fixed-point framework. To overcome this issue, we propose a novel differentiation-based method for handling the $Z$ process. This approach enables improved management of the Lipschitz continuity of decoupling fields, facilitating the well-posedness of the discretized FBSDE system with fully coupled drift. We rigorously prove the convergence of our Markovian iteration method in this more complex setting. Finally, numerical experiments confirm our theoretical insights, showcasing the effectiveness and accuracy of the proposed methodology.

Convergence of the Markovian iteration for coupled FBSDEs via a differentiation approach

TL;DR

The paper tackles solving fully Z-coupled FBSDEs by extending the Markovian iteration with a differentiation-based treatment of the Z-term, enabling control of decoupling-field Lipschitz constants and well-posedness of the discretized scheme. The authors establish uniform Lipschitz and linear-growth bounds for the decoupling fields, derive time-discretization error estimates, and prove convergence of the Markovian iterations under generalized weak/monotone conditions. They also show that enforcing the relation within a single backward regression is crucial for stability, and numerical experiments confirm convergence and accuracy, highlighting advantages over direct extension methods. The results provide a rigorous foundation for solving Z-coupled FBSDEs and support the applicability of related Deep BSDE approaches in this more general setting.

Abstract

In this paper, we investigate the Markovian iteration method for solving coupled forward-backward stochastic differential equations (FBSDEs) featuring a fully coupled forward drift, meaning the drift term explicitly depends on both the forward and backward processes. An FBSDE system typically involves three stochastic processes: the forward process , the backward process representing the solution, and the process corresponding to the scaled derivative of . Prior research by Bender and Zhang (2008) has established convergence results for iterative schemes dealing with -coupled FBSDEs. However, extending these results to equations with coupling poses significant challenges, especially in uniformly controlling the Lipschitz constant of the decoupling fields across iterations and time steps within a fixed-point framework. To overcome this issue, we propose a novel differentiation-based method for handling the process. This approach enables improved management of the Lipschitz continuity of decoupling fields, facilitating the well-posedness of the discretized FBSDE system with fully coupled drift. We rigorously prove the convergence of our Markovian iteration method in this more complex setting. Finally, numerical experiments confirm our theoretical insights, showcasing the effectiveness and accuracy of the proposed methodology.

Paper Structure

This paper contains 12 sections, 15 theorems, 135 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Markovian iteration for FBSDEs with $Z$-coupling
  3. Convergence Analysis
  4. Preliminaries
  5. Lipschitz continuity
  6. Linear growth
  7. Convergence of Markovian iteration
  8. Convergence in time steps
  9. Numerical examples
  10. FBSDE with fully coupled drift
  11. FBSDE with only $Z$-coupling
  12. Acknowledgement

Key Result

Lemma 1

We fix index $i$, and, for $l=1$, 2, let where $X_i^l$ is $\mathcal{F}_{t_i}$-measurable. Assume $\varphi^1$ and $\xi^1$ are uniformly Lipschitz continuous. Then, for any $\lambda_0>0$ and $\lambda_1>0$, we have and in the case that $\varphi^1 = \varphi^2$ and $\xi^1 = \xi^2$, we set $\lambda_0 = \lambda_1 = 0$ to obtain

Figures (2)

  • Figure 1: Convergence results for Example \ref{['example1']}.
  • Figure 2: Convergence results for Example \ref{['example2']}.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • ...and 19 more