Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering

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).