In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is to prove the stable convergence of a sequence of such conditional expectations by using the techniques of martingale limit theorems in the spirit of Jacod (On continuous conditional Gaussian martingales and stable convergence in law, seminaire de probabilites, XXXI, lecture notes in mathematics, Springer, Berlin, 1997).

中文翻译：

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连续时间非线性滤波的欧拉法的渐近误差分布

在本文中，我们针对连续时间观测的非线性滤波问题推导了欧拉方法的渐近误差分布。正如几位作者在以前的工作中所研究的那样，该方法的误差结构的特点是对多个随机积分的某些泛函的条件期望。我们的主要结果是通过使用 Jacod 精神中的鞅极限定理的技术来证明一系列这样的条件期望的稳定收敛（关于连续条件高斯鞅和法律中的稳定收敛，seminaire de probabilites，XXXI，讲义数学，斯普林格，柏林，1997 年）。