Study Of The Solution Of DMZ Equation In Nonlinear Filtering

In order to achieve the property of the dynamical system or control the system,we need to know the state of the system for most engineering applications.However,we can not observe the state directly in general cases,instead we get information of the state from corresponding observations.How to estimate the state under the observation is what the filtering problem considering about.It has huge significance to study the nonlinear filtering problem because nonlinear filtering problem is a main part in real life.Lots of research has been done since the coming of Kalman Filter.People extended Kalman Filter due to its restrict in linear cases and Gaussian assumption.The system can be approximated as a linear system by using Extended Kalman Filter(EKF),Unscented Kalman Filter(UKF),when it is a slightly nonlinear system.One can use the Gausssain sum Filter(GSM)to approximate the Gaussian system.While,these methods don't work well on highly nonlinear system.It is more practicle to consider the conditional density function(cdf)of the state.The stochastic partical differential equation which the cdf is satisfied is called Kushner-FKK equation.We need to solve an infinitesimal system in order to solve this equation.We can not get a closed form solution for it either.We can get the Duncan-Mortensen-Zakai equation which the unnormalized conditional density function satisfied by using Kallianpur-Stribel formula via Girsanov transform.Therefore,Duncan-Mortensen-Zakai equation is a hot research.We can not have explicit solution for this stochastic partial differential equation in most manner,but we can use the PDE method to solve the corresponding robust PDE by using the change of the measure.In this paper,we consider a special filtering system,e.g.the Yau filtering system.The classical Kalman-Bucy filtering system and Benes filtering systems are special cases of Yau filtering system.The observation is degree one polynomial in finite dimensional Yau filtering system in general case.While it is usually nonlinear in actual applications.We study the Yau filtering system with nonlinear observation with linear growth under the condition that the state noise and observation noise is independent.We first give the direct method for the two specific Yau filtering system,eg.Kalman-Bucy fiter and Benes filter.We use the direct method in time interval by using the exponential transform twice.We give a fast and direct method to approximate the Gaussian density.It is shown our Gaussian approximation algorithm needs little time and has high accuracy.The direct method we improved does good estimate in MSE sense.It is more stable than EKF.We can use it in real time and memoryless.For all,our study is worth in solving the actual filtering problems.