Generalised particle filters with Gaussian measures

The stochastic filtering problem deals with the estimation of the posterior distribution of the current state of a signal process X = {X_t}_t≥0 given the information supplied by an associate process Y ={Y_t}_t≥0. The scope and range of its applications includes the control of engineering systems, global data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. A massive scientific and computational effort is dedicated to the development of viable tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions. In higher dimensions, a class of numerical methods called particle filters have proved the most successful methods to-date. These methods produce an approximations of the posterior distribution by using the empirical distribution of a cloud of particles that explore the signal´s state space. We discuss here a more general class of numerical methods which involve generalised particles, that is, particles that evolve through larger spaces. Such generalised particles include Gaussian measures, wavelets, and finite elements in addition to the classical particle methods. We will construct the approximating particle system under the Gaussian measure framework and prove the corresponding convergence result.