In this paper, we address the problem of risk-sensitive filtering and
smoothing for
discrete-time Hidden Markov Models (HMM) with finite-discrete states. The
objective of risk-sensitive filtering is to
minimise the expectation of the exponential of the squared estimation
error weighted by a risk-sensitive parameter. We use the so-called Reference
Probability Method in solving this problem. We achieve finite-dimensional
linear recursions in the information state, and thereby the
state estimate
that minimises the risk-sensitive cost index. Also, fixed-interval smoothing
results are derived.
We show that
$L_2$ or risk-neutral filtering for HMMs can be extracted as a limiting
case of the risk-sensitive filtering problem when the risk-sensitive
parameter approaches zero.
{\bf Key Words:} Hidden Markov model, risk-sensitive filtering, information
state, fixed-interval smoothing.