Some Studies On Regular Zakai Filter For Nonlinear Filtering Problems

The nonlinear filtering models have important applications in various scientific and engineering fields,such as biology,financial mathematics,target tracking and signal processing,etc.The Zakai filter is an effective method for solving nonlinear filtering problems,and the splitting-up algorithm is an important tool for solving the complex differential equation.In this paper,we will combine these two methods to construct numerical scheme for nonlinear filtering models.In existing literatures,many authors apply splitting-up method to decompose the corresponding Zakai equation into a degenerate second-order determined partial differential equation and a second-order stochastic partial differential equation.The degeneracy of the equation makes it difficult to construct effective numerical methods.We overcome the degeneracy by introducing a regularization factor into Zakai equation.Then we construct an effective numerical method and providing error estimates and convergence order analysis.In this paper,we mainly study the Zakai filter for nonlinear filtering model driven by correlated Wiener processes.We first derive the corresponding Zakai equation and use the splitting-up technique to decompose it into a first-order stochastic partial differential equation and a second-order determined partial differential equation,based on which we construct a splitting-up approximate solution.Then,we discretize the time variable using finite difference method to construct a temporal semi-discrete splittingup approximate solution and investigate its convergence order.We also discretize the spatial variable using the spectral Galerkin method to construct a spatial semi-discrete splitting-up approximate solution and analyze its convergence.Combining the splittingup technique,finite difference method and spectral Galerkin method,we construct a fully discrete splitting-up numerical solution and study its convergence and convergence order.In order to improve the computational efficiency,we apply adaptive techniques during the iterations.Finally,we present some numerical experiments to demonstrate the effectiveness of the theoretical analysis.We also study nonlinear filtering model driven by both of Poisson processes and correlated Wiener processes.We manage to derive the corresponding Zakai equation.We first perform regularization transformation to Zakai equation and then use splittingup technique to decompose it into three sub-equations,including a regular stochastic differential equation driven by Wiener process,a regular second-order determined partial differential equation and a regular stochastic differential equation driven by Poisson process.Then we construct a splitting-up approximate solution and prove that it converges to the solution of Zakai equation with 1/2 order accuracy.Combining splitting-up technique,finite difference method and spectral Galerkin method,we respectively construct a temporal semi-discrete splitting-up approximate solution,spatial semi-discrete splitting-up approximate solution and fully discrete splitting-up numerical solution.We investigate convergence and convergence order for these three types of solutions.Finally,we present some numerical experiments to illustrate the theoretical analysis.