A survey of numerical methods for nonlinear filtering problems

The main goal of filtering is to obtain, recursively in time, good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same. In settings where the signal/observation dynamics are significantly nonlinear or the noise intensities are high, an extended Kalman filter (EKF), which is essentially a first order approximation to an infinite dimensional problem, can perform quite poorly: it may require very frequent re-initializations and in some situations may even diverge. The theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations, in the space of probability measures. We survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations; with a special emphasis on an important family of schemes known as the particle filters. A numerical study is presented to illustrate that in settings where the signal/observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended Kalman filter.