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

Paper: arXiv 2504.02814

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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

Cited References (111)

1. An analytic COS method for compound option valuation (2026) — link
2. The compound BSDE method: A fully forward method for option pricing and optimal stopping problems in finance — → 2601.18634
3. Convergence of Markovian iteration for Z-coupled FBSDEs via Gaussian smoothing. — → 2504.02814
4. Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle — → 2401.17472
5. A forward-backward stochastic algorithm for quasi-linear PDEs (2006) — link
6. The steepest descent method for forward-backward SDEs (2005) — link
7. Solving high-dimensional partial differential equations using deep learning (2018) — link
8. Dynamic programming and stochastic control processes (1958) — link

6 of 111 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.