Convergence of the Markovian Iteration for Coupled FBSDEs via a Differentiation Approach

In this paper, we investigate the Markovian iteration method for solving coupled forward-backward stochastic differential equations (FBSDEs) with a fully coupled drift term of the form $b(t,X_t,Y_t,Z_t)$. 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 $Y$. Previous work by Bender and Zhang (2008) established convergence results for iterative schemes for $Y$-coupled FBSDEs. However, extending these results to equations with $Z$ coupling presents significant challenges, particularly in obtaining a uniform control of the Lipschitz constants 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 better control of the Lipschitz constants 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, we develop an efficient algorithm for computing the resulting numerical scheme, and numerical experiments confirm the theoretical findings and demonstrate the effectiveness and accuracy of the proposed methodology.