Symplectic Runge-Kutta methods for the Kalman-Bucy filter

In this paper, numerical methods for the Kalman-Bucy filter are investigated from the viewpoint of geometry. The differential matrix Riccati equation for the Kalman-Bucy filter is transformed into a linear differential Hamiltonian system. We show that the linear differential Hamiltonian system with two different initial conditions is on symplectic group. The two different initial conditions relate to two different statistical assumptions about the initial state of a linear time-varying dynamical system. Then, symplectic Runge-Kutta methods can be applied to the linear differential Hamiltonian system, which keep the numerical solution on the symplectic group. Numerical examples are given to illustrate the performance of the numerical methods.