Estimating the state in stiff continuous-time stochastic systems within extended Kalman filtering

SIAM Journal on Scientific Computing , 38(6) (2016), A3565-A3588
http://dx.doi.org/10.1137/15M1039833

This paper studies performance of the well-known extended Kalman filtering method for estimating continuous-time stochastic mathematical models with discrete measurements. The mentioned continuous-discrete stochastic systems arise naturally in applied science and engineering and constitute the basic simulation means in such diverse topics of research as target tracking, navigation, stochastic control, chemistry, finance and so on. Therefore developing effective tools for their numerical treatment is an important task of study nowadays. Our paper focuses on so-called stiff stochastic systems in the sense that the stiffness of the drift coefficient in stochastic differential equations feeds into severe problems with state estimation of such systems by existing extended Kalman filtering methods. We expose that the majority of filters are not effective for stiff stochastic systems, including those attributed to solving such mathematical models. So, the task of designing efficient methods for treating stiff continuous-time stochastic systems with discrete measurements naturally arises in applied science and engineering. Here, we present two promising state estimators grounded in the nested implicit Runge--Kutta method of order 6. To increase the accuracy and robustness of state estimation, our methods are supplied with automatic error control mechanisms. The new filters are examined numerically on the stochastic Van der Pol oscillator, which can expose a stiff behavior, and compared to other techniques designed for estimating continuous-time stochastic models.