Convergence of Markovian Iteration for $Z$-Coupled FBSDEs via Gaussian Smoothing

Zhipeng Huang and Cornelis W. Oosterlee

Papers from arXiv.org

Abstract: In this paper, we investigate the Markovian iteration method for coupled forward-backward stochastic differential equations (FBSDEs) with drift $b(t,X_t,Y_t,Z_t)$ and deterministic, time-dependent diffusion. Bender and Zhang (2008) established convergence results for Markovian iteration in the $Y$-coupled setting. Extending these results to equations with $Z$-coupling presents additional challenges, particularly in obtaining uniform control of the spatial Lipschitz constants of the discrete decoupling fields across iterations and time steps. We address this difficulty by representing the discrete $Z$-field through the spatial derivative of the Gaussian-smoothed $Y$-field. This representation, combined with weak differentiation, yields uniform spatial Lipschitz bounds for both discrete decoupling fields without requiring their classical second derivatives. Under suitable contraction conditions, we prove convergence of the Markovian iteration and well-posedness of the discrete scheme. With additional regularity assumptions, we derive an overall error bound accounting for both iteration and time discretization. Numerical experiments compare three practical implementations and demonstrate the effectiveness of the derivative-based and merged approaches.

Date: 2025-04, Revised 2026-10
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