Estimation of the Quadratic Variation of Nearly Observed Semimartingales with Application to Filtering

Abstract

Consider a filtering problem in which the available information is a noisy observation of a continuous semimartingale $H_t $. In the case of a high signal-to-noise ratio, it is proved that $H_t $ and its quadratic variation can be jointly estimated by means of a finite-dimensional filter; moreover, for this result, the observation noise and $H_t $ are not required to be independent. This problem can be viewed as a linear filtering problem with randomly time-varying parameters, and our filter is auto-adaptive with respect to changes of the parameters. These results are then applied to the nonlinear filtering of Markov diffusion processes when the observation function is not injective but satisfies a weaker detectability assumption. It appears that filtering such a system involves two timescales. The study is based on time discretization; the main tools are an averaging principle and an application of the asymptotic ordinary differential equation method for the study of stochastic algorithms.