A Splitting Method For Nonlinear Filtering Problem With Correlation Noise

In this paper,we study how to solve the nonlinear filtering problem with correlated noise.First,we transform the problem to a new measure space through measure transfor-mation.Then we solve the Zakai equation satisfied by the non-standard density function of the solution of the filtering problem.And we use the numerical method to solve the problem.In time,we use the splitting method and the Euler iterative method.In space,we use the Galerkin method,and the convergence of the splitting algorithm is analyzed and proved.Finally,the convergence of the numerical method is verified by numerical experiments.This paper is divided into five parts.In the introduction part,we analyse the develop-ment background of the filtering problem firstly,as well as the main ideas,advantages and disadvantages of various solutions,and then introduces the main problems to be solved and the structure of the article.In the second part,the filtering model with correlated noise is introduced,and the Zakai equation satisfied by the nonstandard density function in the new measure space is derived,as well as the existence and uniqueness of its solution.In the third part,how to apply the splitting algorithm and Galerkin method to solving Zakai equation is given,and the prior estimation of the splitting algorithm is obtained.In the fourth part,we prove and obtain the convergence of the splitting algorithm and the order of convergence is 1.In the fifth part,we use the difference method to discretize the equations obtained by the splitting algorithm,and prove that the convergence order of the discretization method is 1/2.Finally,the convergence order of the algorithm is verified by numerical experiments.