We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the approximate random variable $X^\varepsilon_t$ converges to the original random variable $X_t$ in total variation distance and we estimate the error. We also give an estimate of the distance between the densities of the laws of the two random variables. These are done by using some integration by parts techniques in Malliavin calculus.

翻译：我们用跳跃处理随机差分方程。 为了获得准确的近似方案, 通常用布朗动议取代“ 小型跳跃 ” 。 在本文中, 我们证明, 每一个固定时间, 大约随机变量$X ⁇ varepsilon_ t$, 与原始随机变量$X_ t$总变差距离一致, 我们估计出错误。 我们还给出了两个随机变量法律密度之间的距离估计。 这些是使用马利亚文微积分中部分技术的集成完成的 。