\section{Conclusion}
The problem of discrete-time filtering and smoothing with an exponential
quadratic error cost-criteria, termed risk-sensitive filtering and smoothing, has
been addressed in this paper using a reference probability method.
A new probability measure has been defined where observations are {\em i.i.d} and the reformulated cost-criteria has been minimised to give
filtering and smoothing results for continuous-range nonlinear and linear
state space models and Hidden Markov Models with finite-discrete state.
Closed form results for the optimising estimate
and the density of the smoothed estimate have been given in the case of continuous-range non-linear
signal models (nonlinearity with respect to both the state process and
noise). Infinite-dimensional linear recursions have been obtained for the
information state for both nonlinear and linear state space models. For the
linear discrete-time state space model, explicit analytical results have been
obtained for filtering and smoothing. Results for risk-sensitive filtering and smoothing
have been also obtained for Hidden Markov Models with finite discrete-state
processes where recursions for the information state
have been shown to be linear and finite-dimensional.
These new results suggest a strong duality between risk-sensitive control and
risk-sensitive filtering. This duality is explored in a subsequent work, along with a risk-sensitive dual
control is problem.