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

ArXiv ID: 2504.02814 “View on arXiv”

Authors: Unknown

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.

Keywords: forward-backward stochastic differential equations (FBSDEs), Markovian iteration method, decoupling fields, Lipschitz continuity, fixed-point framework, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced stochastic calculus, Lipschitz analysis, and convergence proofs for coupled FBSDEs, while the empirical section provides only theoretical numerical examples without real-world data or extensive backtesting metrics.

  flowchart TD
    A["Research Goal<br/>Convergence of Markovian Iteration<br/>for Coupled FBSDEs with Z-Drift"] --> B["Key Methodology<br/>Differentiation Approach for Z Process"]
    B --> C["Computational Process<br/>Discretized Fixed-Point Framework"]
    C --> D{"Iterative Solving<br/>Lipschitz Controlled?"}
    D -- Yes --> E["Key Findings<br/>Convergence Proven &<br/>Numerical Validation"]
    D -- No --> C