关于具有有界光滑系数的SDE边际分布的难解求积问题
On hard quadrature problems for marginal distributions of SDEs with bounded smooth coefficients
AI总结：
本文研究了具有无限次可微且有界系数的SDE的求积问题，证明了任何基于有限系数函数值的近似方法在最坏情况下都无法保证误差收敛速度。
摘要：
在Hairer等人最近的工作中，构造了一个具有无限次可微且有界系数的随机微分方程（SDE），使得Euler方法在近似最终时间解的第一个分量的期望值时收敛但无法达到多项式速率的均方误差。本文证明，这种性能不佳并非Euler方案的特定缺陷，而是在任何基于有限系数函数值的近似方法在最坏情况下均会观察到。更糟糕的是，对于任何基于有限次序列评估系数及其偏导数的蒙特卡洛方法，以及任意慢的收敛速度，存在一个具有无限次可微且有界系数的SDE序列，其中所有扩散系数的一阶导数被限制为1，而所有漂移系数的一阶导数被单个实值函数统一支配，使得对应的近似误差序列无法比给定的速度更快收敛到零。
英文摘要：
In recent work of Hairer, Hutzenthaler and Jentzen, see [9], a stochastic differential equation (SDE) with infinitely often differentiable and bounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In the present paper we show that this type of bad performance for quadrature of SDEs with infinitely often differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely often differentiable and bounded by one coefficients such that the first order derivatives of all diffusion coefficients are bounded by one as well and the first order derivatives of all drift coefficients are uniformly dominated by a single real-valued function and such that the corresponding sequence of mean absolute errors for approximation of the expected value of the first component of the solution at the final time can not converge to zero faster than the given speed.
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