论文标题
L2与无限系数的随机微分方程平滑近似的L2收敛
L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients
论文作者
论文摘要
本文的目的是在随机微分方程（SDES）的分段线性近似值中获得$ C^1 $漂移和$ C^2 $扩散系数，具有均匀界限的衍生物。对此类Wong-Zakai近似值的收敛分析最常假设SDE的系数均匀界定。尽管$ l^q $收敛似乎尚未确定，并且对于涉及蒙特卡洛近似值的几种应用，但几乎可以使用现在的标准粗糙路径技术在无限情况下肯定会收敛。我们认为在无限制的情况下，使用传统随机分析和粗糙的路径技术来考虑$ l^2 $融合。我们期望我们的证明技术扩展到更通用的分段平滑近似。
The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although $L^q$ convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider $L^2$ convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.
