On the convergence rates of a general class of weak approximations of SDEs
On the convergence rates of a general class of weak approximations of SDEs
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关于一类SDE弱近似的收敛速度
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
S.Ghazali
中科院分区：
文献类型：
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作者：
D. Crisan;S.Ghazali
摘要本文给出了一类随机微分方程解的弱逼近的收敛性分析。这类包括最近的近似，如Kusuoka的时刻相似的家庭方法和维也纳空间的方法的LyonsVictoir立方。我们表明，收敛速度本质上取决于所选择的测试函数的光滑性。对于光滑函数（所需的光滑度取决于近似的阶数），在寻求近似的时间间隔上进行等距划分是最佳的。对于不太光滑的函数（例如Lipschitz函数），收敛速度会下降，最佳划分不再是等距的。我们的分析依赖于Kusuoka-Stroock关于随机微分方程解的分布的光滑性的结果。最后将所得结果应用于滤波问题的数值求解。
Abstract In this paper, the convergence analysis of a class of weak approximations of solutions of stochastic differential equations is presented. This class includes recent approximations such as Kusuoka’s moment similar families method and the LyonsVictoir cubature of Wiener Space approach. We show that the rate of convergence depends intrinsically on the smoothness of the chosen test function. For smooth functions (the required degree of smoothness depends on the order of the approximation), an equidistant partition of the time interval on which the approximation is sought is optimal. For functions that are less smooth (for example Lipschitz functions), the rate of convergence decays and the optimal partition is no longer equidistant. Our analysis rests upon Kusuoka-Stroock’s results on the smoothness of the distribution of the solution of a stochastic differential equation. Finally the results are applied to the numerical solution of the filtering problem.