Smooth particle filters for likelihood evaluation and maximisation

In this paper,a method is introduced for approximating the likelihood for the unknown parameters of a state space model.The approximation converges to the true likelihood as the simulation size goes to infinity. In addition,the approximating likelihood is continuous as a function of the unknown parameters under rather general conditions.The approach advocated is fast, robust and avoids many of the pitfalls associated with current techniques based upon importance sampling.We assess the performance of the method by considering a linear state space model, comparing the results with the Kalman filter, which delivers the true likelihood. We also apply the method to a non-Gaussian state space model, the Stochastic Volatility model, finding that the approach is efficient and effective. Applications to continuous time finance models are also considered. A result is established which allows the likelihood to be estimated quickly and efficiently using the output from the general auxiliary particle filter.