论文标题
完全耦合的McKean-Vlasov前向后SDES的后验错误估计
A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEs
论文作者
论文摘要
完全耦合的McKean-Vlasov向前的随机微分方程（MV-FBSDE）自然来自大种群优化问题。通常很难判断MV-FBSDE的给定数值解决方案的质量，通常需要PICARD迭代和嵌套条件期望的近似值。本文提出了A后验误差估计器，以量化在时间网格上任意生成的近似值的$ l^2 $ approximation误差。我们确定误差估计器等于给定的数值解和正向Euler离散化MV-FBSDE之间的全局近似误差。分析中的关键和挑战性的步骤是该Euler近似与MV-FBSDE的稳定性证明，这是独立的。我们进一步证明，对于足够精细的时间网格，还可以通过误差估计器来测量求解连续MV-FBSDE的数值解决方案的精度。误差估计证明使用基于残差的算法解决MV-FBSDE是合理的。由平均场控制和游戏引起的MV-FBSDE的数值实验证实了误差估计器的有效性和实际适用性。
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.
