Optimal linear stochastic estimation theory, which is known as Kalman filtering theory, has been dominant for the past two decades. In application to Gauss-Markov systems, it
achieves the conditional mean estimate, being at the same time
the minimum variance estimate and indeed also the maximum-likelihood estimate \cite{Moore}.
The term minimum variance estimate implies the minimization of
the energy of the estimation error, or the squared filtering error. Of course,
minimum variance estimation can be achieved also for nonlinear stochastic
systems via infinite-dimensional filters in general. A more
general estimation problem is minimising the exponential of the squared filtering error, or its expectation, thus penalizing
all the higher order moments of the
estimation error energy.
This problem is termed the {\em risk-sensitive filtering} problem, in analogy with a corresponding risk-sensitive control problem. Risk-sensitive filtering also makes connection with the so-called $H_{\infty}$ filtering problem. The index of the
exponential is usually weighted by a risk-sensitive parameter
which exaggerates the error when the risk is high due to plant and noise
uncertainty, so that risk-sensitive filtering allows a trade-off between
optimal filtering for the nominal model case and the average noise
situation, and robustness to worst case noise and model uncertainty. When
this risk-sensitive parameter approaches zero, the filter is the optimal
$L_2$ filter, termed here {\em risk-neutral filter}.
The risk-sensitive filtering problem has been addressed for linear Gauss-Markov signal models in \cite{Speyer}.
The optimising estimate is derived from a linear filter. In fact, it is an $H_{\infty}$ filter.
Off-line Riccati equations are solved to achieve the filter gain which becomes the so-called Kalman gain when the risk-senstive parameter approaches
zero. Risk-sensitive control problems are relatively more abundant in literature
\cite{Whittle} \cite{van} \cite{Jacob}. Recently a solution to the output feedback problem
for linear and nonlinear discrete-time systems using information state techniques has been proposed
in \cite{JBE} \cite{Iain}. Also,
tracking problems for the linear, exponential, quadratic index case have been
solved in \cite{Iain}. The feedback and feedforward gains for the information state in this case require the solution of a backwards Riccati and linear equation, analogous to the standard Linear Quadratic Gaussian (LQG) tracking problem solution. The derivation techniques are based on a reference probability method.
The risk-sensitive filtering problem is similar in nature
to its control counterpart, and it makes sense to ask whether there are corresponding nonlinear stochastic risk-sensitive filtering results, perhaps dualizing
the risk-sensitive control results. Instead of solving a backward
dynamic programming to obtain a sequence of admissible controls, we can, at
each time point,
calculate the filtered estimates recursively in the forward direction based on
the observations available to that point. This is a more natural approach than the backward dynamic programming approach taken in
\cite{Speyer}.
Risk-sensitive filtering problems are closely connected with $H_{\infty}$ filtering theory developed in \cite{Shaked} \cite{Uchita}. A solution to the
nonlinear risk-sensitive problem, therefore is expected to give a direct
solution to the nonlinear $H_{\infty}$ filtering problem.
In this paper, we consider quite general stochastic nonlinear state-space signal models, involving information states, and derive, in the first instance,
information state filters based on the risk-sensitive cost index. These filters
are linear and infinite-dimensional.
The optimising estimate is then given
as the minimising argument of a particular integral, which is of course,
infinite-dimensional. More specifically, the linear
Gauss-Markov model is treated as a special case, and the same results
as in \cite{Speyer} are obtained. In addition, Hidden Markov Models
with finite-discrete states are considered. Backward filters and fixed-interval smoothing
results are given for all these various signal models.
The derivation techniques used in
this paper are different than the ones used for earlier filtering results in \cite{Speyer} but similar to
those used for the control results in \cite{Iain}. This measure change technique has been proposed and developed in
\cite{MElliot}. It is based on Girsanov's Theorem, Kolmogorov's Extension Theorem, and Fubini's Theorem. The preliminary task is to define a new probability measure
where the observations are independently identically distributed ({\em i.i.d}).
Then, one can reformulate the
optimisation problem in the new measure to obtain the recursions in the
information state, the expression for the optimising filtered estimate, and also density functions of the smoothed estimates by using and exploiting the
independence of the observations. Solving the problem in the new measure is
equivalent to solving the problem in the old measure as long as a restriction
is set on a certain Radon-Nikodym derivative described. Moreover, it is shown that known risk-neutral
filtering results can be recovered from the risk-sensitive results as a special case when the risk-sensitive parameter tends to zero.
In Sec. 2, we describe a nonlinear stochastic state space model, formally define the risk-sensitive filtering
problem, and then deal with the change of measure and reformulation of the problem in the new probability measure to achieve the
filtering and smoothing results. In Sec. 3,
we derive the risk-sensitive filters and smoothers for
a Hidden Markov Model with finite-discrete states. Sec. 4 specialises the
results of Sec. 2 to achieve linear risk-sensitive filters and smoothers.
In Sec. 5, we introduce
a more general nonlinear signal model and briefly talk
about the measure change technique and state the filtering and
smoothing results. Sec. 6. presents the connection between risk-neutral and
risk-sensitive filtering and Sec. 7 presents some concluding remarks.
In the Appendix, we provide proofs for the two theorems
concerning the linear stochastic state-space model.