Uniform Stability of a Particle Approximation of the Optimal Filter Derivative

Particle methods, also known as Sequential Monte Carlo methods, are a principled set of algorithms used to approximate numerically the optimal filter in nonlinear non-Gaussian statespace models. However, when performing maximum likelihood parameter inference in state-space models, it is also necessary to approximate the derivative of the optimal filter with respect to the parameter of the model. References [G. Poyiadjis, A. Doucet, and S. S. Singh, Particle methods for optimal filter derivative: Application to parameter estimation, in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 5, Philadelphia, 2005, pp. 925– 928 and G. Poyiadjis, A. Doucet, and S. S. Singh, Biometrika, 98 (2011), pp. 65–80] present an original particle method to approximate this derivative, and it was shown in numerical examples to be numerically stable in the sense that it did not deteriorate over time. In this paper we theoretically substantiate this claim. Lp bounds and a central limit theorem for this particle approximation are presented. Under mixing conditions these Lp bounds and the asymptotic variance are uniformly bounded with respect to the time index.