© The Institution of Electrical Engineers
The paper describes a recently developed reformulation of the optimalfiltering equations for a noisily observed diffusion process and discusses the computational implications. The computation involved is the solution of a parabolic partial differential equation whose coefficients are determined by the observed process. A Monte Carlo method of solution is proposed and given in detail as an example. It is argued that nonlinear filtering is now a practical proposition.
References


1)

R.S. Bucy ,
P.D. Joseph
.
(1968)
, Filtering for stochastic processes with applications to guidance.

2)

Bucy, R.S., Hecht, C., Senne, K.D.: `An engineer's guide to building nonlinear filters', report SRLTR720004, 1972.

3)

R.S. Bucy ,
K.D. Senne
.
Nonlinear filtering algorithms for vector processing machines.
Comput. & Math. with Appl.
,
317 
338

4)

J.M.C. Clark ,
J.K. Skwirzynski
.
(1978)
The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication systems and random process theory.

5)

M.H.A. Davis
.
(1977)
, Linear estimation and stochastic control.

6)

M.H.A. Davis ,
M. Hazewinkel ,
J.C. Willems
.
(1981)
Pathwise nonlinear filtering, Stochastic systems the mathematics of filtering and identification and applications.

7)

M.H.A. Davis
.
On a multiplicative functional transformation arising in nonlinear filtering theory.
Z. Wahrscheinlichkeitstheorie ver Geb.
,
125 
139

8)

M.H.A. Davis ,
S.I. Marcus ,
M. Hazewinkel ,
J.C. Willems
.
(1981)
An introduction to nonlinear filtering, Stochastic systems the mathematics of filtering and identification and applications.

9)

M.H.A. Davis ,
P.H. Wellings
.
(1980)
Computational problems in nonlinear filtering, Analysis and Optimization of Systems.

10)

G.B. di Masi ,
W. Runggaldier
.
Continuoustime approximations to the nonlinear filtering problem.
App. Math. & Optimiz.

11)

M. Fujisaki ,
G. Kallianpur ,
H. Kunita
.
Stochastic differential equations for the nonlinear filtering problem.
Osaka J. Math.
,
19 
40

12)

A. Gelb
.
(1975)
, Applied optimal estimation.

13)

J.M. Hammersley ,
D.C. Handscomb
.
(1964)
, Monte Carlo methods.

14)

M. Hazewinkel ,
J.C. Willems
.
(1981)
, Stochastic systems, the mathematics of filtering and identification and applications.

15)

G. Kallianpur
.
(1980)
, Stochastic filtering theory.

16)

A.J. Krener
.
KalmanBucy and minimax filtering.
IEEE Trans.
,
291 
292

17)

H. Kunita
.
Densities of measurevalued process governed by a stochastic partial differential equation.
Syst. & Control Lett.

18)

H.J. Kushner
.
On the differential equations satisfied by the conditional probability densities of Markov processes.
SIAM J. Control
,
106 
119

19)

H.J. Kushner
.
(1977)
, Probability methods for approximations in stochastic control and for elliptic equations.

20)

H.J. Kushner
.
A robust discretestate approximation to the optimal nonlinear filter for a difussion.
Stochastics
,
75 
83

21)

R.S. Liptser ,
A.N. Shiryaev
.
(1977)
, Statistics of random processes I.

22)

S.K. Mitter ,
M. Hazewinkel ,
J.C. Willems
.
(1981)
Lectures on nonlinear filtering and stochastic mechanics, Stochastic systems the mathematics of filtering and identification and applications.

23)

E.J. McShane
.
(1974)
, Stochastic calculus and stochastic models.

24)

E. Pardoux ,
M. Hazewinkel ,
J.C. Willems
.
(1981)
Nonlinear filtering, prediction and smoothing, Stochastic systems the mathematics of filtering and identification and applications.

25)

E. Pardoux
.
Equations du filtrage nonlinéaire, de la prediction et du lissage.
Stochastics

26)

G. Strang ,
G.J. Fix
.
(1973)
, An analysis of the finite element method.

27)

E. Wong
.
(1971)
, Stochastic processes in information and dynamical systems.
http://iet.metastore.ingenta.com/content/journals/10.1049/ipd.1981.0037
Related content
content/journals/10.1049/ipd.1981.0037
pub_keyword,iet_inspecKeyword,pub_concept
6
6