Particle methods are popular computational tools for Bayesian inference in nonlinear non-Gaussian state space models. For this class of models, we present two particle algorithms to compute the score vector and observed information matrix recursively. The first algorithm is implemented with computational complexity O(N ) and the second with complexity O(N 2 ), where N is the number of particles. Although cheaper, the performance of the O(N ) method degrades quickly, as it relies on the approximation of a sequence of probability distributions whose dimension increases linearly with time. In particular, even under strong mixing assumptions, the variance of the estimates computed with the O(N ) method increases at least quadratically in time. The more expensive O(N 2 ) method relies on a nonstandard particle implementation and does not suffer from this rapid degradation. It is shown how both methods can be used to perform batch and recursive parameter estimation.

In this paper we address the problem of on-line estimation of unknown static parameters in non-linear nonGaussian state-space models. We consider a particle filtering method and employ two gradient-free Stochastic Approximation (SA) methods to maximize recursively the likelihood function, the Finite Difference SA and Spall's Simultaneous Perturbation SA. We demonstrate how these algorithms can generate maximum likelihood estimates in a simple and computationally efficient manner. The performance of the proposed algorithms is assessed through simulation.

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This paper proposes novel particle methods for online parameter estimation for partially observed diffusions. We consider diffusions observed with error under a non-linear mapping and multivariate diffusions where only a subset of the components is observed. The proposed methods rely on the commonly used idea of data augmentation and are based on obtaining particle approximations to the derivatives of the optimal filter. The performance of our algorithms is assessed using several financial applications.