  Sequential Monte Carlo methods, also known as particle methods, are a widely
used set of computational tools for inference in non-linear non-Gaussian
state-space models. In many applications it may be necessary to compute the
sensitivity, or derivative, of the optimal filter with respect to the static
parameters of the state-space model; for instance, in order to obtain maximum
likelihood model parameters of interest, or to compute the optimal controller
in an optimal control problem. In Poyiadjis et al. [2011] an original particle
algorithm to compute the filter derivative was proposed and it was shown using
numerical examples that the particle estimate was numerically stable in the
sense that it did not deteriorate over time. In this paper we substantiate this
claim with a detailed theoretical study. Lp bounds and a central limit theorem
for this particle approximation of the filter derivative are presented. It is
further shown that under mixing conditions these Lp bounds and the asymptotic
variance characterized by the central limit theorem are uniformly bounded with
respect to the time index. We demon- strate the performance predicted by theory
with several numerical examples. We also use the particle approximation of the
filter derivative to perform online maximum likelihood parameter estimation for
a stochastic volatility model.
