Fully coupled McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the $L^2$-approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator.

翻译：完全耦合的McKean-Vlasov正倒向随机微分方程（MV-FBSDEs）自然产生于大规模群体优化问题中。评估MV-FBSDEs给定数值解的质量通常很困难，因为这通常需要Picard迭代和嵌套条件期望的近似。本文提出了一种后验误差估计器，用于量化时间网格上任意生成近似的$L^2$逼近误差。我们证明了该误差估计器与给定数值解和前向欧拉离散化MV-FBSDE解之间的全局逼近误差等价。分析中一个关键且具有挑战性的步骤是证明该欧拉近似对MV-FBSDE的稳定性，这一结论本身也具有独立意义。我们进一步证明，对于足够精细的时间网格，求解连续MV-FBSDE的数值解的精度也可通过该误差估计器度量。这些误差估计为求解MV-FBSDE的残差型算法提供了理论依据。针对平均场控制和博弈问题中MV-FBSDE的数值实验，验证了该误差估计器的有效性与实际适用性。