This thesis is concerned with recursive Bayesian estimation of non-linear dynamical

systems, which can be modeled as discretely observed stochastic differential

equations. The recursive real-time estimation algorithms for these continuous-

discrete filtering problems are traditionally called optimal filters and the algorithms

for recursively computing the estimates based on batches of observations

are called optimal smoothers. In this thesis, new practical algorithms for approximate

and asymptotically optimal continuous-discrete filtering and smoothing are

presented.

The mathematical approach of this thesis is probabilistic and the estimation

algorithms are formulated in terms of Bayesian inference. This means that the

unknown parameters, the unknown functions and the physical noise processes are

treated as random processes in the same joint probability space. The Bayesian approach

provides a consistent way of computing the optimal filtering and smoothing

estimates, which are optimal given the model assumptions and a consistent

way of analyzing their uncertainties.

The formal equations of the optimal Bayesian continuous-discrete filtering

and smoothing solutions are well known, but the exact analytical solutions are

available only for linear Gaussian models and for a few other restricted special

cases. The main contributions of this thesis are to show how the recently developed

discrete-time unscented Kalman filter, particle filter, and the corresponding

smoothers can be applied in the continuous-discrete setting. The equations for the

continuous-time unscented Kalman-Bucy filter are also derived.

The estimation performance of the new filters and smoothers is tested using

simulated data. Continuous-discrete filtering based solutions are also presented to

the problems of tracking an unknown number of targets, estimating the spread of

an infectious disease and to prediction of an unknown time series.