Filtering and Stochastic Control
Contributed by Ljubo Vlacic.
A comprehensive survey of linear filtering theory can be found in 
- H. W. Bode and C. E. Shannon proposed the solution to the
problem of prediction and smoothing (). A modern
account of the solution can be found in  and
more detailed treatment of the ideas are presented in 
- R. E. Kalman (,,) made explicit that an effective solution to the
Wiener-Hopf equation using method of spectral factorization () could be obtained when the continuous process
had a rational spectral density.
- Stratanovich derived the conditional density equation using
the so-called Stratanovich calculus ().
- The theory of optimal stochastic control in the fully
observable case is quite similar to that of non-linear filtering in
connection with the linear quadratic stochastic control problem (). Early works in this area are due to Howard (), Florentin (), and
Fleming (); See also .
- Inspired by the development of Dynamic Programming by Bellman
() and the ideas of Caratheodory ()
related to Hamilton-Jacobi Theory, the development of optimal control
of nonlinear dynamical systems took place (, ), see , ,
 for further details of the ideas.
- The solution to quadratic cost optimal control for linear
stochastic dynamical systems was provided by Florentin (,
), by Joseph in discrete-time (), and by Kushner (). The
definitive treatment of the problem was proposed by Wonham (), see also .
- The partially observable stochastic control problem treated
by Florentin (), Davis and Varayia () and Fleming and Pardouz ().
Detailed discussions can be found in  and the
- For a good discussion on the distinction between open-loop
stochastic control and feedback control see .
- Non-linear filters are almost always infinite dimensional and
there are only a few known examples where the filter is known to be a
finite dimension. The Kalman filter is an example and the other
finite-state cases are first discussed in   and also  .
- A difficulty is that one of the fundamental equations of
non-linear filtering turns out to be a non-linear stochastic partial
differential equation (). Zakai (),
Duncan (), and Mortensen ()
proposed alternative solutions to the above difficulty which involves a
linear stochastic differential equation.
- Giransov introduced the idea of measure transformation in
stochastic differential equation, see , ,  and the references
therein for details.
- The earlier ideas of nonlinear filtering were developed and
introduced by Forest and Kailath (), and in
definitive form by Fujisaki, Kallianpur, Kunita ().
- Bobrovsky and Zakai proposed a method for obtaining lower
bounds on the mean-squared error ().
- As an attempt to address some of the issues with non-linear
filtering, pathwise non-linear filtering was considered where the
filter depends continuously on the output (, ).
- The Linear Quadratic Gaussian methodology and optimal
non-linear stochastic control have found a wide variety of applications
in aerospace, multi-variable control design systems, finance, etc. (, ).