Filtering systems of coupled stochastic
differential equations partially observed at high frequency
We introduce particle filters for sequential estimation of unobserved components of systems of partially observed stochastic differential equations (SDEs) which have a state-independent volatility and a drift function which is of gradient form. The method allows the efficient simultaneous simulation of unobserved states given batches of data by expressing the filtering distribution as a change of measure from a simulation smoother. The proposed filter has desirable robustness properties and it is particularly appropriate for high frequency observations. The method is based on novel extensions of the exact algorithm for simulation and inference of diffusions, and the filters do not need to introduce any approximations through time-discretisation of the process. Additionally, we introduce a new methodology for sequential Monte Carlo based on unbiased but not necessarily positive estimators of the importance sampling weights. We demonstrate our methods on simulated data.