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Filtering for a Signal Given by a Linear Stochastic Retarded Differential Equation
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
212, 75]87 Ž1997.
AY975454
Filtering for a Signal Given by a Linear Stochastic Retarded Differential Equation S. A. Elsanousi Department of Mathematics, College of Science, King Saud Uni¨ ersity, P.O. Box 2455, Riyadh, 11451, Saudi Arabia Submitted by Sandy L. Zabell Received May 4, 1995
A filtering of Kalman]Bucy type is derived for a signal governed by a linear retarded stochastic differential equation, given a noisy observation process linearly related to the section of the signal. A Volterra type integral equation is obtained for a ‘‘general tracking error.’’ Q 1997 Academic Press
1. INTRODUCTION The filtering problem is concerned with a pair of random processes x Ž t ., y Ž t .. Of these x Ž t ., called the signal process, is not observed directly and y Ž t ., called the observation process, is a noisy function of x Ž t .. The problem is to determine at each time t, ˆ x Ž t . [ E x Ž t . N y Ž u. ;0 F u F t 4 , the optimal estimate for x Ž t . in the mean square sense given knowledge of the observed process up to time t. The problem has been extensively studied; excellent exposes ´ can be found in, for example, Liptser and Shirayev w8x, Kallianpur w6x, Oksendal w11x, and Elliot w4x. For fairly general processes x Ž t ., y Ž t . and f in a wide class H of real valued functions on the state space of x Ž t ., one can determine a stochastic differential relation satisfied by the process Zˆf Ž t . [ E f Ž x Ž t .. N y Ž u. ;0 F u F t 4 . The relation is in general not recursive Žunless the totality of relations ; f g H is construed as a measure-valued stochastic integral equation.. A notable exception is the Kalman]Bucy filtering problem where x Ž t ., y Ž t . are given by dx Ž t . s F Ž t . x Ž t . dt q C Ž t . dU Ž t . ,
x Ž 0. s x 0
Ž 1.1.
dy Ž t . s G Ž t . x Ž t . dt q D Ž t . dV Ž t . ,
y Ž 0. s 0
Ž 1.2.
75 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
76
S. A. ELSANOUSI
with UŽ t ., V Ž t . mutually independent Brownian motions and x 0 a Gaussian variable independent of both. In this case ˆ x Ž t . is given by a linear stochastic differential equation parametrised by the tracking error E< x Ž t . y ˆ x Ž t .< 2 4 , which itself is determined by a Riccati equation. In the present work we seek a recursive relation for ˆ x Ž t . in the case where x Ž t . is given by a linear retarded stochastic differential equation and where the observable process depends linearly on the section x t Žto be defined below.. Not unexpectedly, perhaps, it turns out that we need to take on board E x t N y Ž u., 0 F u F t 4 . It was only after this manuscript was prepared for publication that the author became aware of the work of Kolmanovskii and Shaikhet w7x, where this problem was handled for discrete delays through variational methods. Our approach covers the case of non-discrete delays as well and is different in that it consists in modifying the innovations approach for the Kalman]Bucy problem using the relevant facts about retarded equations as presented in the comprehensive work of Mohammed w10x.
2. NOTATION AND ASSUMPTIONS Let Ž V, F , P . denote a complete probability space on which are defined all stochastic processes under consideration. J is the interval wyr, 0x Žwith r G 0.. C Ž J, R n . is the Banach space of all continuous paths h : J ª R n with the sup norm 5h 5 C s sup s g J
ht Ž s . s h Ž t q s . . I is the interval w0, T x Žwith T ) 0.. UŽ t .: t g I 4 and V Ž t .: t g I 4 are two mutually independent Brownian motions with respective dimensions p and q. c Ž s .: s g J 4 is a Gaussian process with Holder continuous paths ¨ independent of both UŽ t .4t g I and V Ž t .4t g I . It is assumed that c g L 2 Ž V, F, P, C Ž J, R n ... For all t g I, we let AŽ t ., F Ž t . denote respectively the s-algebras generated by c Ž s ., UŽ u.: s g J, 0 F u F t 4 and by c Ž s ., UŽ u., V Ž ¨ .: s g J, 0 F u, ¨ F t 4 . By augumenting if necessary we assume each F Ž t . to contain all null sets. We note that F Ž t . is independent of UqŽ t . and VqŽ t ., the s-algebras generated by the increments UŽ t 2 . y UŽ t 1 ., V Ž t 2 . y V Ž t 1 . for t 2 ) t 1 G t. C: I ª LŽR p , R n . and D: I ª LŽR q, R m . are bounded measurable. Furthermore we assume that the smallest eigenvalues of the matrices DŽ t . DŽT .* Ž t g I . are uniformly bounded away from zero Ž* denotes
FILTERING FOR A RETARDED EQUATION
77
transposition.. Consequently the inverses w DŽ t . DŽ t .*xy1 exist and are uniformly bounded. Furthermore for each t g I, there exists a symmetric matrix K Ž t . such that K Ž t . 2 s w DŽ t . DŽ t .*xy1 . F: I = C Ž J, R n . ª R n and G: I = C Ž J, R n . ª R m are measurable maps such that ; t g I, F Ž t, .. and GŽ t, .. are bounded linear. Furthermore we assume that each of the sets 5 F Ž t, ..5: t g I 4 and 5 GŽ t, ..5: t g I 4 is bounded. If B: C Ž J, R n . ª R l is any bounded linear map, we shall retain the same symbol B for the extension of B to C˜Ž J, R n ., the space of bounded measurable paths Žvia the Riesz representation theorem., and for the map induced by B Žcolumnwise. on C Ž J, LŽR k , R n ... We trust this slight abuse of notation will not cause any confusion. Should any process in the sequel possess a continuous version, it will tacitly be assumed that this version has already been selected. The symbol s shall in the text be reserved exclusively for members of J; the symbols u, ¨ , t refer to members of I.
3. THE FILTERING EQUATIONS Let the signal process x Ž t . be given by dx Ž t . s F Ž t , x t . dt q C Ž t . dU Ž t .
Ž 3.1.
x0 s c
Ž 3.2.
and let the observation process y Ž t . be given by dy Ž t . s G Ž t , x t . dt q D Ž t . dV Ž t . y Ž 0 . s 0.
Ž 3.3. Ž 3.4.
Denote by Y Ž t . the s-algebra generated by y Ž u. ;0 F u F t, and if Ž z Ž t .. t g I is any stochastic process, set where appropriate
ˆz Ž t . s E z Ž t . N Y Ž t . 4
and
zŽ t. s zŽ t. y ˆ zŽ t. .
Our aim is to determine ˆ x Ž t .. Although we shall eventually state our results for the multidimensional case, we shall in the interest of clarity, detail the derivation of the filtering equations only in the case m s n s p s q s 1. C Ž J . shall stand for C Ž J, R..
78
S. A. ELSANOUSI
If 0 F u F t and h g L 2 Ž V, F Ž0., P, C Ž J .., let Stuh be the section of the solution of Ž3.1. starting at u from h , i.e.,
¡ F Ž ¨ , S h . d¨ q H ¢h Ž t q s y u. , tqs
Ž
Stuh
h Ž 0. q H . Ž s . s~ u
tqs
u
¨
C Ž u . dU Ž u . ,
u
tqsGu t q s - u.
Ž 3.5. Likewise, let for h g C Ž J ., Ttuh be the section of the solution of the deterministic equation dz Ž t . s F Ž t , z t . dt
Ž 3.6.
starting at u from h ,
¡ F Ž ¨ , T h . d¨ , ¢h Ž t q s y u. , tqs
Ž Ttuh . Ž s . s
~h Ž 0. q Hu
u
¨
tqsGu
Ž 3.7.
t q s - u.
We make the following observations Žcf. w10x.: Ž0.1. Stuh is a well defined member of L 2 Ž V, F Ž t ., P, C Ž J .. whenever h g L 2 Ž V, F Ž0., P, C Ž J ... Stu can be extended to L 2 Ž V, F Ž0., P, C˜Ž J ... The extension shall still be denoted by Stu ; this is the basic existence and uniqueness theorem. Ž0.2. Stu is Lipschitz; in fact there exist constants K, M such that 5 Stuh1 y Stuh 2 5 L 2 Ž V , C . F Ke M Ž tyu. 5h1 y h 2 5 L 2 Ž V , C . . Ž0.3. For all h g L 2 Ž V, F Ž0., P, C Ž J .. the map t ¬ Ttuh is continuous on I. If h has Holder continuous paths, then the boundedness ¨ conditions on F and C ensure that the process Ž s, w . ¬ Ž Stuh .Ž s .Ž v . has Holder continuous paths. ¨ Ž0.4. Besides the validity of the above observations for Ttu ŽTtu is S tu when C Ž t . ' 0., we note that Ttu is linear on C Ž J ... THEOREM 1.
If h g C Ž J ., then ;0 F u F t Stuh s Ttuh q
t
¨
Hu T DC Ž ¨ . dU Ž ¨ . , t
Ž 3.8.
79
FILTERING FOR A RETARDED EQUATION
where D is the indicator function of 04 on J. The integral on the right hand side is understood as the map of J gi¨ en by t
¨
Hu T DC Ž ¨ . dU Ž ¨ . t
t
Ž s . s H Tt¨DC Ž ¨ . Ž s . dU Ž ¨ . . u
Proof. This is a simple modification of w10, Theorem ŽVI, 4.1.x, where the autonomous case is handled. The proof consists in showing that
¡ ¢h Ž t y u. ,
t
Ž T h . Ž 0. q H y Ž t . s~ u u t
Tt¨DC Ž ¨ . Ž 0 . dU Ž ¨ . ,
tGu t-u
is a solution of Ž3.1. starting at u from h , and has section yt equal to the right hand side of Ž3.8.. COROLLARY 1. If x Ž t . s Ž St0c .Ž0., then x Žt .: t g wyr, T x4 is a Gaussian system. Furthermore, x Žt . is AŽt . measurable ;t g I and AŽ0. measurable if t g J. Proof. St0c Ž 0 . s Tt0c Ž 0 . q H0t Tt¨DC Ž ¨ . Ž 0 . dU Ž ¨ . . Since w Tt¨DC Ž ¨ .xŽ0. is deterministic, the random field H0t w Tt¨DC Ž u.xŽ0. dUŽ ¨ .: t g I 4 is a Gaussian system. Since evaluation at 0 and Tt0 are continuous linear, there exists a Borel measure m t such that Tt0h Ž 0 . s
0
Hyr h Ž s . d m Ž s . t
;h g C Ž J . .
Ž 3.9.
If P n s s0 , s1 , . . . , s2 k 4 is the partition of J into 2 k equal parts, then 2 ky1
0
lim Ý Hyr c Ž s . d m Ž s . s kª` t
c Ž si . m t si , siq1 .
Ž a.s..
is0
so that if z s Ýljs1 l j w Tt0j c xŽ0. is any linear combination in w Tt0c xŽ0.: t g w0, T x4 , we have z s lim
kª`
2 ky1
Ý is1
l
c Ž si .
Ý
l j m t j si , siq1 . ,
js1
a normally distributed random variable. Thus ŽTt0c .Ž0.: t g I 4 is also a Gaussian system. Furthermore, the representation Ž3.9. confirms that Tt0c
80
S. A. ELSANOUSI
is AŽ0. measurable ; t g I. Independence of AŽ0. and UqŽ 0. guarantees that x Žt .: t g wyr, T x4 is indeed Gaussian. The remaining statement is already obvious. COROLLARY 2. Let x Ž t . be the solution of Ž3.1., Ž3.2. and y Ž t . that of Ž3.3., Ž3.4.. Then the composite system x Žt .: t g wyr, T x4 j y Žt .: t g I 4 is Gaussian. Proof. Consider any linear combination k
zs
l
Ý
li x Ž t i . q
is1
Ý
mj y Ž tj . ,
t j g I.
js1
Then zs
½
k
l
li x Ž t i . q
Ý is1
Ý
mj
js1
tj
H0
5 ½
G Ž u, x u . du q
l
Ý
mj
js1
tj
H0
5
D Ž u . dV Ž u . .
Corollary 1 and the fact that D is deterministic indicate that each braced term is Gaussian; their independence confirms that z is also Gaussian. In what follows, let x Ž t . be the solution of Ž3.1., Ž3.2. and write X Ž t . for x t . It is important to note that XˆŽ t . is not the section ˆ x t of the map t ¬ˆ x Ž t .. Assuming for the time being that EŽ c . s 0, we have from Theorem 1 that E x Ž t . 4 s E St0c Ž 0 . 4 s E Ž Tt0c . Ž 0 . 4 s Tt0 E c 4 Ž 0 . s 0
;t g I
and the way is paved to quote the following well known consequences of Corollary 2 Žsee w6, 8x and in particular w11x whose notation we follow closely.: ŽF1 . For all t, ¨ g I, E x Ž t . N Y¨ 4 s P¨ x Ž t . where P¨ is the projection of L V, F , P, R. onto L Ž y, ¨ ., the closed linear hull of y Ž u.: 0 F u F ¨ 4 . ŽF2 . If B Ž t . is the innovation process defined by 2Ž
BŽ t. s
t
H0
G Ž u, X Ž u . . D Ž u.
du q V Ž t .
then Ž B Ž t .. t g I coincides with 1-dimensional Brownian motion. Furthermore L Ž y, ¨ . s L Ž B, ¨ . s
¨
½H
0
f Ž u . dB Ž u . : f g L 2 w 0, ¨ x .
5
81
FILTERING FOR A RETARDED EQUATION
Since X Ž t . g L 2 Ž V, C Ž J .., it follows that E 5 X Ž t .GŽ t, X Ž t ..5 C - ` and E 5 XˆŽ t .GŽ t, XˆŽ t ..5 C - `. Consequently X Ž t .GŽ t, X Ž t .. and XˆŽ t .GŽ t, XˆŽ t .. are Bochner integrable and we can define g Ž t . in C Ž J . by
g Ž t . s E X Ž t . G Ž t , X Ž t . . 4 y E XˆŽ t . G Ž t , XˆŽ t . . 4 .
Ž 3.10.
We can think of g Ž t . as a generalized tracking error. Indeed, in the special case where GŽ t, h . s GŽ t .h Ž0., g Ž t . takes the form g Ž t . s GŽ t . E N x Ž t . y ˆ x Ž t .< 2 4 . Observe also that
g Ž t . s E X Ž t . GŽ t , X Ž t . . 4 . THEOREM 2.
Ž 3.11.
For all t g I, ;s g J, XˆŽ t . Ž s . s
Ttug Ž u . Ž s .
t
H0
D Ž u.
dB Ž u . .
Ž 3.12.
Proof. Due to facts ŽF1 ., ŽF2 ., ' g s g L 2 w0, t x such that XˆŽ t . Ž s . s E x Ž t q s . N Y Ž t . 4 s
t
H0 g Ž u . dB Ž u . s
a.s.
Since I y Pt x Ž t q s . is orthogonal to L Ž B, t ., we have E x Ž t q s . B Ž u . 4 s ²Pt x Ž t q s . q w I y Pt x x Ž t q s . , B Ž u . : s ²Pt x Ž t q s . , B Ž u . : sE s
t
½H
0
u
H0
g s Ž ¨ . dB Ž ¨ . ?
t
H0 x
w0, ux
Ž ¨ . dB Ž u .
5
g s Ž ¨ . d¨ .
Also since x Ž t q s . and V Ž t . are independent Žand hence orthogonal., E x Ž t q s . . B Ž u. 4 s
u
H0
E x Ž t q s . .G Ž ¨ , X Ž ¨ . . DŽ ¨ .
d¨
and consequently g s Ž u. s
E x Ž t q s . .G Ž u, X Ž u . . 4 D Ž u.
Ž a.a.u . .
Ž 3.13.
82
S. A. ELSANOUSI
Now, t
x Ž t q s . s Ttu X Ž u . Ž s . q
Hu
TtuDC Ž ¨ . Ž s . dU Ž ¨ . .
Since GŽ u, X Ž u.. is F Ž u. measurable, and F Ž u. is independent of Uq Ž u., we have on using the linearity of Ttu and evaluation at s E x Ž t q s . .G Ž u, X Ž u . . 4 s E Ttu X Ž u . Ž s . .G Ž u, X Ž u . . 4 s Ttu E X Ž u . .G Ž u, X Ž u . . 4 Ž s . s Ttug Ž u . Ž s . . Therefore we have from Ž3.13. XˆŽ t . Ž s . s
t
H0
Ttug Ž u . Ž s . D Ž u.
dB Ž u .
Ž a.s.. .
Remark. The stipulated boundedness conditions on F, G, C and the Holder continuity of c entail that the process Ž s, w . ¬ w Ttug Ž u.xŽ s, v . has ¨ Holder continuous paths. An application of Kolmogorov’s theorem then ¨ ensures the existence of a continuous version of the process Ž s, v . ¬ w H0t ŽTtug Ž u.rDŽ u..Ž s . dBŽ u.xŽ v .. If this version is selected, we will then have the stronger result: XˆŽ t . s
; t g I,
t
H0
Ttug Ž u . D Ž u.
dB Ž u . Ž in C Ž J . . a.s.
Ž 3.14.
Since dBŽ u. s Ž dy Ž u. y GŽ u, XˆŽ u.. du.rDŽ u., we can write Ž3.14. in the form t XˆŽ t . s H 0
Ttug Ž u . D2 Ž u.
dy Ž u . y
t
H0
Ttug Ž u . .G Ž u, XˆŽ u . . D2 Ž u.
du. Ž 3.15.
In order to obtain XˆŽ t . Žand thereby ˆ x Ž t . s XˆŽ t .Ž0.., we need to determine g Ž t . first. By linearity of GŽ t, .., we have
g Ž t . Ž s . s G Ž t , E X Ž t . . X Ž t . Ž s . 4 . y G Ž t , E XˆŽ t . . XˆŽ t . Ž s . 4 . From Theorem 1 X Ž t . Ž s . s Tt0c Ž s . q
t
H0
TtuDC Ž u . Ž s . dU Ž u . .
;s g J.
83
FILTERING FOR A RETARDED EQUATION
Since Tt0c is AŽ0. measurable, and AŽ0. is independent of UqŽ 0., we have ;s g J E X Ž t . Ž s . . X Ž t . Ž s . 4 s E Tt0c . Ž s . . Ž Tt0c . Ž s . 4 t
q
H0
TtuDC Ž u . Ž s . . TtuDC Ž u . Ž s . du.
Since from Theorem 2 XˆŽ t . Ž s . s
Ttug Ž u .
t
H0
Ž s . dB Ž u . ,
D Ž u.
we have E XˆŽ t . Ž s . . XˆŽ t . Ž s . 4 s
t
H0
Ttug Ž u . Ž s . . Ttug Ž u . Ž s . D2 Ž u.
du.
Consequently, we deduce that
g Ž t . Ž s . s G Ž t , E Tt0c . T0t c Ž s . 4 . q G t ,
ž
ž
y G t,
t
H0
Ttug Ž u . . Ttug Ž u . Ž s . D 2 Ž u.
s E G Ž t , Tt0c . Tt0c Ž s . 4 q y
t
H0
ž
G t,
u
u
t
du
t
/
u
u
t
/
Ž s . du
t
H0 G Ž t , T DC Ž u . . T DC Ž u .
Ttug Ž u . . Ttug Ž u . Ž s . D 2 Ž u.
t
H0 T DC Ž u . . T DC Ž u .
t
Ž s . . du
du.
If we define H: I = C Ž J . ª C Ž J . by H Ž t , h . Ž s . s G Ž t , h .h Ž s . . we may write
g Ž t . s E H Ž t , Tt0c . 4 q
t
H0
H Ž t , TtuDC Ž u . . du y
t
H0
ž
H t,
Ttug Ž u . D Ž u.
/
du.
Ž 3.16.
/
84
S. A. ELSANOUSI
By considering the signal x Ž t . y E x Ž t .4 and the observation y Ž t . y H0t GŽ u, E x u4. du, we can relax the requirement EŽ c . s 0 and get the filtering equations XˆŽ t . s E Tt0c 4 q t
y
H0
D 2 Ž u.
Ttug Ž u . D 2 Ž u.
g Ž t . s E H Ž t , Tt0 c y E Ž c . Tt g Ž u .
dy Ž u .
.G Ž u, XˆŽ u . . du
Ž 3.159.
t
. 4 q H H Ž t , TtuDC Ž u . . du 0
u
t
y
Ttug Ž u .
t
H0
H0 H
ž
t,
D Ž u.
/
du.
Ž 3.169.
Finally, to get these equations in the multi-dimensional case, we begin by choosing a symmetric matrix K Ž t . such that K Ž t . 2 s w DŽ t . DŽ t .*xy1 for each t g I. We then set BŽ t. s
t
t
H0 G Ž u, X Ž u . . du q H0 K Ž u . D Ž u . dV Ž u .
g Ž t . s E X Ž t . .G Ž t , X Ž t . . * 4 y E XˆŽ t . .G Ž t , XˆŽ t . . * 4
Ž 3.17. Ž 3.18.
and modify the derivation of Ž3.159., Ž3.169. with no particular difficulties to get THEOREM 3. Gi¨ en the assumptions layed out in Section 2, the solution of the filtering equations Ž3.1. ] Ž3.4. is ˆ x Ž t . s XˆŽ t .Ž0. where XˆŽ t . s E Ž Tt0c . q t
t
H0 T g Ž u . .
H0 T g Ž u . .
y
0
t
u
t
D Ž u. D Ž u. * y1
D Ž u. D Ž u. *
y1
dy Ž u .
G Ž u, XˆŽ u . . du
Ž 3.19.
and
g Ž t . s E H Ž t , Tt0 c y E Ž c . t
t
. 4 q H H Ž t , TtuDC Ž u . . du 0
H0 H Ž t , T g Ž u . . K Ž u . . du.
y
u
t
Ž 3.20.
85
FILTERING FOR A RETARDED EQUATION
Here we ha¨ e H: I = C Ž J, R n . ª C Ž J, LŽR m , R n .. gi¨ en by H Ž t , h . Ž s . s G Ž t , h .h Ž s . * . *.
Ž 3.21.
Remarks. Ž1. Equation Ž3.20. is a non-linear Volterra integral equation with phase space C Ž J, R n .. One suspects that iterative methods for its solution can be obtained along the lines of those for equations with a finite dimensional phase space. Equation Ž3.19. is a stochastic Volterra type integral equation in C Ž J, R n .. If we imbed C Ž J, R n . in L2 Ž J, R n ., then equations of this type afford solutions via the Picard iteration procedure Žcf. w3x.. Ž2. If GŽ t, h . s GŽ t . g Žh . for some m = m matrix GŽ t . and a bounded linear g: C Ž J, R n . ª R m we can write
g Ž t . s b Ž t . G* Ž t . , where b Ž t . s E X Ž t .. g Ž X Ž t ..*4 y E XˆŽ t .. g Ž XˆŽ t ..*4 . An obvious modification of the above argument leads to the equations XˆŽ t . s E Tt0c 4 q t
t
H0 T b Ž u . .G* Ž u . . u
t
H0 T b Ž u . .G* Ž u . .
y
u
t
D Ž u. D Ž u. *
b Ž t . s E h Ž Tt0 c y E Ž c . t
y1
D Ž u. D Ž u. * y1
dy Ž u .
G Ž u . g Ž XˆŽ u . . du Ž 3.22.
t
. 4 q H h Ž TtuDC Ž u . . du 0
H0 h Ž T b Ž u . .G* Ž u . K Ž u . . du,
y
u
Ž 3.23.
t
where hŽh .Ž s . s w g Žh.h *Ž s ..x* ;h g C Ž J, R n ., s g J. In the special case when g s e0 , the evaluation at 0, we get t
ˆx Ž t . s E Ž Tt0c . Ž 0 . 4 q H Ttub Ž u . Ž 0. .G* Ž u . . D Ž u . D Ž u . *
y1
dy Ž u .
0
t
H0 Ž T b . Ž u . Ž 0. .G* Ž u .
y
u
t
D Ž u. D Ž u. *
y1
GŽ u. ˆ x Ž u . du Ž 3.24.
which is a stochastic integral equation with configuration space L 2 Ž V, R n .; and so we would not need to solve for XˆŽ t . in L 2 Ž V, C Ž J, R n .. to obtain the solution ˆ x Ž t . of the filtering problem. The presence of XˆŽ u. in the right hand side of Ž3.19. and Ž3.22. precludes such possibilities even in the
86
S. A. ELSANOUSI
case of fixed delay: g Žh . s h Žyr .. Observe that XˆŽ u.Žyr . s E x Ž u y r . N Y Ž u.4 may not equal ˆ x Ž u y r . s E x Ž u y r . N Y Ž u y r .4 . Ž3. If we set r s 0, then C Ž J, R n . s R n, F Ž t, h . s F Ž t ..h , GŽ t, h . s GŽ t ..h for some bounded matrices F Ž t ., GŽ t ., DC Ž t . s C Ž t ., X Ž t . s x Ž t ., and Ž r t .Ttuh s F Ž t .ŽTtuh .. Using the formula for differentiation under the stochastic integral Žcf. w11, Lemma VI, 4.4x., we recapture the Kalman]Bucy equations viz. dxˆŽ t . s F Ž t . y b Ž t . G* Ž t . D Ž t . D Ž t . * q b Ž t . G* Ž t . D Ž t . D Ž t . * db dt
y1
y1
GŽ t . 4 ˆ x Ž t . dt
dy Ž t .
s FŽ t. b Ž t. q b Ž t. FŽ t.* q CŽ t. CŽ t.* y b Ž t . GŽ t . * DŽ t . DŽ t . *
y1
b *Ž t . GŽ t . .
Ž4. The imposition of boundedness Žrather than square integrability . on the coefficients C, D, F, G and of Holder continuity on the paths of c ¨ Žrather than mere continuity. is motivated by our desire for the process Ž s, v . ¬ w H0t Žw Ttug Ž u.xrDŽ u..xŽ s . dBŽ u.xŽ v . to possess a continuous version so that Ž3.14. would hold in C Ž J .. We do not know in general if continuity of f Ž u, ? . ; u would guarantee the continuity of a version of Ž s, v . ¬ H0t f Ž u, x . dBŽ u.. ACKNOWLEDGMENT The author thanks S. E. A. Mohammed of Southern Illinois University for suggesting the problem in the first place and for some very valuable discussions.
REFERENCES 1. C. T. H. Baker, ‘‘The Numerical Treatment of Integral Equations,’’ Clarendon Press, Oxford, 1977. 2. A. T. Bharucha-Reid, ‘‘Random Integral Equations,’’ Academic Press, New YorkrLondon, 1972. 3. P. L. Chow, Stochastic partial differential equations in turbulence related problems, in ‘‘Probabilistic Analysis and Related Topics,’’ Vol. I, Academic Press, New YorkrSan FranciscorLondon, 1978. 4. J. Elliot, Stochastic calculus and applications, in ‘‘Applications of Mathematics,’’ Vol. 18, Springer-Verlag, New YorkrHeidelbergrBerlin, 1982. 5. J. Hale, Theory of functional differential equations, in ‘‘Applied Mathematical Sciences,’’ Vol. 3, Springer-Verlag, New YorkrHeidelbergrBerlin, 1977.
FILTERING FOR A RETARDED EQUATION
87
6. G. Kallianpur, Stochastic filtering theory, in ‘‘Applications of Mathematics,’’ Vol. 13, Springer-Verlag, New YorkrHeidelbergrBerlin, 1980. 7. V. B. Kolmanovskii and L. E. Shaikhet, State estimate of hereditary stochastic systems, Stochastic Anal. Appl. 7, No. 4 Ž1989., 387]411. 8. R. S. Liptser and A. N. Shirayev, Statistics of random processes, I, in ‘‘Applications of Mathematics,’’ Vol. 5, Springer-Verlag, New YorkrHeidelbergrBerlin, 1982. 9. M. Metivier and J. Pellaumail, ‘‘Stochastic Integration,’’ Academic Press, LondonrNew York, 1980. 10. S. E. A. Mohammed, Stochastic functional differential equations, in ‘‘Pitman Research Notes in Mathematics,’’ Vol. 99, Pitman, BostonrLondonrMelbourne, 1984. 11. B. Oksendal, ‘‘Stochastic Differential Equations: An Introduction with Applications,’’ Springer-Verlag, BerlinrHeidelbergrNew YorkrTokyo, 1985.
212, 75]87 Ž1997.
AY975454
Filtering for a Signal Given by a Linear Stochastic Retarded Differential Equation S. A. Elsanousi Department of Mathematics, College of Science, King Saud Uni¨ ersity, P.O. Box 2455, Riyadh, 11451, Saudi Arabia Submitted by Sandy L. Zabell Received May 4, 1995
A filtering of Kalman]Bucy type is derived for a signal governed by a linear retarded stochastic differential equation, given a noisy observation process linearly related to the section of the signal. A Volterra type integral equation is obtained for a ‘‘general tracking error.’’ Q 1997 Academic Press
1. INTRODUCTION The filtering problem is concerned with a pair of random processes x Ž t ., y Ž t .. Of these x Ž t ., called the signal process, is not observed directly and y Ž t ., called the observation process, is a noisy function of x Ž t .. The problem is to determine at each time t, ˆ x Ž t . [ E x Ž t . N y Ž u. ;0 F u F t 4 , the optimal estimate for x Ž t . in the mean square sense given knowledge of the observed process up to time t. The problem has been extensively studied; excellent exposes ´ can be found in, for example, Liptser and Shirayev w8x, Kallianpur w6x, Oksendal w11x, and Elliot w4x. For fairly general processes x Ž t ., y Ž t . and f in a wide class H of real valued functions on the state space of x Ž t ., one can determine a stochastic differential relation satisfied by the process Zˆf Ž t . [ E f Ž x Ž t .. N y Ž u. ;0 F u F t 4 . The relation is in general not recursive Žunless the totality of relations ; f g H is construed as a measure-valued stochastic integral equation.. A notable exception is the Kalman]Bucy filtering problem where x Ž t ., y Ž t . are given by dx Ž t . s F Ž t . x Ž t . dt q C Ž t . dU Ž t . ,
x Ž 0. s x 0
Ž 1.1.
dy Ž t . s G Ž t . x Ž t . dt q D Ž t . dV Ž t . ,
y Ž 0. s 0
Ž 1.2.
75 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
76
S. A. ELSANOUSI
with UŽ t ., V Ž t . mutually independent Brownian motions and x 0 a Gaussian variable independent of both. In this case ˆ x Ž t . is given by a linear stochastic differential equation parametrised by the tracking error E< x Ž t . y ˆ x Ž t .< 2 4 , which itself is determined by a Riccati equation. In the present work we seek a recursive relation for ˆ x Ž t . in the case where x Ž t . is given by a linear retarded stochastic differential equation and where the observable process depends linearly on the section x t Žto be defined below.. Not unexpectedly, perhaps, it turns out that we need to take on board E x t N y Ž u., 0 F u F t 4 . It was only after this manuscript was prepared for publication that the author became aware of the work of Kolmanovskii and Shaikhet w7x, where this problem was handled for discrete delays through variational methods. Our approach covers the case of non-discrete delays as well and is different in that it consists in modifying the innovations approach for the Kalman]Bucy problem using the relevant facts about retarded equations as presented in the comprehensive work of Mohammed w10x.
2. NOTATION AND ASSUMPTIONS Let Ž V, F , P . denote a complete probability space on which are defined all stochastic processes under consideration. J is the interval wyr, 0x Žwith r G 0.. C Ž J, R n . is the Banach space of all continuous paths h : J ª R n with the sup norm 5h 5 C s sup s g J
ht Ž s . s h Ž t q s . . I is the interval w0, T x Žwith T ) 0.. UŽ t .: t g I 4 and V Ž t .: t g I 4 are two mutually independent Brownian motions with respective dimensions p and q. c Ž s .: s g J 4 is a Gaussian process with Holder continuous paths ¨ independent of both UŽ t .4t g I and V Ž t .4t g I . It is assumed that c g L 2 Ž V, F, P, C Ž J, R n ... For all t g I, we let AŽ t ., F Ž t . denote respectively the s-algebras generated by c Ž s ., UŽ u.: s g J, 0 F u F t 4 and by c Ž s ., UŽ u., V Ž ¨ .: s g J, 0 F u, ¨ F t 4 . By augumenting if necessary we assume each F Ž t . to contain all null sets. We note that F Ž t . is independent of UqŽ t . and VqŽ t ., the s-algebras generated by the increments UŽ t 2 . y UŽ t 1 ., V Ž t 2 . y V Ž t 1 . for t 2 ) t 1 G t. C: I ª LŽR p , R n . and D: I ª LŽR q, R m . are bounded measurable. Furthermore we assume that the smallest eigenvalues of the matrices DŽ t . DŽT .* Ž t g I . are uniformly bounded away from zero Ž* denotes
FILTERING FOR A RETARDED EQUATION
77
transposition.. Consequently the inverses w DŽ t . DŽ t .*xy1 exist and are uniformly bounded. Furthermore for each t g I, there exists a symmetric matrix K Ž t . such that K Ž t . 2 s w DŽ t . DŽ t .*xy1 . F: I = C Ž J, R n . ª R n and G: I = C Ž J, R n . ª R m are measurable maps such that ; t g I, F Ž t, .. and GŽ t, .. are bounded linear. Furthermore we assume that each of the sets 5 F Ž t, ..5: t g I 4 and 5 GŽ t, ..5: t g I 4 is bounded. If B: C Ž J, R n . ª R l is any bounded linear map, we shall retain the same symbol B for the extension of B to C˜Ž J, R n ., the space of bounded measurable paths Žvia the Riesz representation theorem., and for the map induced by B Žcolumnwise. on C Ž J, LŽR k , R n ... We trust this slight abuse of notation will not cause any confusion. Should any process in the sequel possess a continuous version, it will tacitly be assumed that this version has already been selected. The symbol s shall in the text be reserved exclusively for members of J; the symbols u, ¨ , t refer to members of I.
3. THE FILTERING EQUATIONS Let the signal process x Ž t . be given by dx Ž t . s F Ž t , x t . dt q C Ž t . dU Ž t .
Ž 3.1.
x0 s c
Ž 3.2.
and let the observation process y Ž t . be given by dy Ž t . s G Ž t , x t . dt q D Ž t . dV Ž t . y Ž 0 . s 0.
Ž 3.3. Ž 3.4.
Denote by Y Ž t . the s-algebra generated by y Ž u. ;0 F u F t, and if Ž z Ž t .. t g I is any stochastic process, set where appropriate
ˆz Ž t . s E z Ž t . N Y Ž t . 4
and
zŽ t. s zŽ t. y ˆ zŽ t. .
Our aim is to determine ˆ x Ž t .. Although we shall eventually state our results for the multidimensional case, we shall in the interest of clarity, detail the derivation of the filtering equations only in the case m s n s p s q s 1. C Ž J . shall stand for C Ž J, R..
78
S. A. ELSANOUSI
If 0 F u F t and h g L 2 Ž V, F Ž0., P, C Ž J .., let Stuh be the section of the solution of Ž3.1. starting at u from h , i.e.,
¡ F Ž ¨ , S h . d¨ q H ¢h Ž t q s y u. , tqs
Ž
Stuh
h Ž 0. q H . Ž s . s~ u
tqs
u
¨
C Ž u . dU Ž u . ,
u
tqsGu t q s - u.
Ž 3.5. Likewise, let for h g C Ž J ., Ttuh be the section of the solution of the deterministic equation dz Ž t . s F Ž t , z t . dt
Ž 3.6.
starting at u from h ,
¡ F Ž ¨ , T h . d¨ , ¢h Ž t q s y u. , tqs
Ž Ttuh . Ž s . s
~h Ž 0. q Hu
u
¨
tqsGu
Ž 3.7.
t q s - u.
We make the following observations Žcf. w10x.: Ž0.1. Stuh is a well defined member of L 2 Ž V, F Ž t ., P, C Ž J .. whenever h g L 2 Ž V, F Ž0., P, C Ž J ... Stu can be extended to L 2 Ž V, F Ž0., P, C˜Ž J ... The extension shall still be denoted by Stu ; this is the basic existence and uniqueness theorem. Ž0.2. Stu is Lipschitz; in fact there exist constants K, M such that 5 Stuh1 y Stuh 2 5 L 2 Ž V , C . F Ke M Ž tyu. 5h1 y h 2 5 L 2 Ž V , C . . Ž0.3. For all h g L 2 Ž V, F Ž0., P, C Ž J .. the map t ¬ Ttuh is continuous on I. If h has Holder continuous paths, then the boundedness ¨ conditions on F and C ensure that the process Ž s, w . ¬ Ž Stuh .Ž s .Ž v . has Holder continuous paths. ¨ Ž0.4. Besides the validity of the above observations for Ttu ŽTtu is S tu when C Ž t . ' 0., we note that Ttu is linear on C Ž J ... THEOREM 1.
If h g C Ž J ., then ;0 F u F t Stuh s Ttuh q
t
¨
Hu T DC Ž ¨ . dU Ž ¨ . , t
Ž 3.8.
79
FILTERING FOR A RETARDED EQUATION
where D is the indicator function of 04 on J. The integral on the right hand side is understood as the map of J gi¨ en by t
¨
Hu T DC Ž ¨ . dU Ž ¨ . t
t
Ž s . s H Tt¨DC Ž ¨ . Ž s . dU Ž ¨ . . u
Proof. This is a simple modification of w10, Theorem ŽVI, 4.1.x, where the autonomous case is handled. The proof consists in showing that
¡ ¢h Ž t y u. ,
t
Ž T h . Ž 0. q H y Ž t . s~ u u t
Tt¨DC Ž ¨ . Ž 0 . dU Ž ¨ . ,
tGu t-u
is a solution of Ž3.1. starting at u from h , and has section yt equal to the right hand side of Ž3.8.. COROLLARY 1. If x Ž t . s Ž St0c .Ž0., then x Žt .: t g wyr, T x4 is a Gaussian system. Furthermore, x Žt . is AŽt . measurable ;t g I and AŽ0. measurable if t g J. Proof. St0c Ž 0 . s Tt0c Ž 0 . q H0t Tt¨DC Ž ¨ . Ž 0 . dU Ž ¨ . . Since w Tt¨DC Ž ¨ .xŽ0. is deterministic, the random field H0t w Tt¨DC Ž u.xŽ0. dUŽ ¨ .: t g I 4 is a Gaussian system. Since evaluation at 0 and Tt0 are continuous linear, there exists a Borel measure m t such that Tt0h Ž 0 . s
0
Hyr h Ž s . d m Ž s . t
;h g C Ž J . .
Ž 3.9.
If P n s s0 , s1 , . . . , s2 k 4 is the partition of J into 2 k equal parts, then 2 ky1
0
lim Ý Hyr c Ž s . d m Ž s . s kª` t
c Ž si . m t si , siq1 .
Ž a.s..
is0
so that if z s Ýljs1 l j w Tt0j c xŽ0. is any linear combination in w Tt0c xŽ0.: t g w0, T x4 , we have z s lim
kª`
2 ky1
Ý is1
l
c Ž si .
Ý
l j m t j si , siq1 . ,
js1
a normally distributed random variable. Thus ŽTt0c .Ž0.: t g I 4 is also a Gaussian system. Furthermore, the representation Ž3.9. confirms that Tt0c
80
S. A. ELSANOUSI
is AŽ0. measurable ; t g I. Independence of AŽ0. and UqŽ 0. guarantees that x Žt .: t g wyr, T x4 is indeed Gaussian. The remaining statement is already obvious. COROLLARY 2. Let x Ž t . be the solution of Ž3.1., Ž3.2. and y Ž t . that of Ž3.3., Ž3.4.. Then the composite system x Žt .: t g wyr, T x4 j y Žt .: t g I 4 is Gaussian. Proof. Consider any linear combination k
zs
l
Ý
li x Ž t i . q
is1
Ý
mj y Ž tj . ,
t j g I.
js1
Then zs
½
k
l
li x Ž t i . q
Ý is1
Ý
mj
js1
tj
H0
5 ½
G Ž u, x u . du q
l
Ý
mj
js1
tj
H0
5
D Ž u . dV Ž u . .
Corollary 1 and the fact that D is deterministic indicate that each braced term is Gaussian; their independence confirms that z is also Gaussian. In what follows, let x Ž t . be the solution of Ž3.1., Ž3.2. and write X Ž t . for x t . It is important to note that XˆŽ t . is not the section ˆ x t of the map t ¬ˆ x Ž t .. Assuming for the time being that EŽ c . s 0, we have from Theorem 1 that E x Ž t . 4 s E St0c Ž 0 . 4 s E Ž Tt0c . Ž 0 . 4 s Tt0 E c 4 Ž 0 . s 0
;t g I
and the way is paved to quote the following well known consequences of Corollary 2 Žsee w6, 8x and in particular w11x whose notation we follow closely.: ŽF1 . For all t, ¨ g I, E x Ž t . N Y¨ 4 s P¨ x Ž t . where P¨ is the projection of L V, F , P, R. onto L Ž y, ¨ ., the closed linear hull of y Ž u.: 0 F u F ¨ 4 . ŽF2 . If B Ž t . is the innovation process defined by 2Ž
BŽ t. s
t
H0
G Ž u, X Ž u . . D Ž u.
du q V Ž t .
then Ž B Ž t .. t g I coincides with 1-dimensional Brownian motion. Furthermore L Ž y, ¨ . s L Ž B, ¨ . s
¨
½H
0
f Ž u . dB Ž u . : f g L 2 w 0, ¨ x .
5
81
FILTERING FOR A RETARDED EQUATION
Since X Ž t . g L 2 Ž V, C Ž J .., it follows that E 5 X Ž t .GŽ t, X Ž t ..5 C - ` and E 5 XˆŽ t .GŽ t, XˆŽ t ..5 C - `. Consequently X Ž t .GŽ t, X Ž t .. and XˆŽ t .GŽ t, XˆŽ t .. are Bochner integrable and we can define g Ž t . in C Ž J . by
g Ž t . s E X Ž t . G Ž t , X Ž t . . 4 y E XˆŽ t . G Ž t , XˆŽ t . . 4 .
Ž 3.10.
We can think of g Ž t . as a generalized tracking error. Indeed, in the special case where GŽ t, h . s GŽ t .h Ž0., g Ž t . takes the form g Ž t . s GŽ t . E N x Ž t . y ˆ x Ž t .< 2 4 . Observe also that
g Ž t . s E X Ž t . GŽ t , X Ž t . . 4 . THEOREM 2.
Ž 3.11.
For all t g I, ;s g J, XˆŽ t . Ž s . s
Ttug Ž u . Ž s .
t
H0
D Ž u.
dB Ž u . .
Ž 3.12.
Proof. Due to facts ŽF1 ., ŽF2 ., ' g s g L 2 w0, t x such that XˆŽ t . Ž s . s E x Ž t q s . N Y Ž t . 4 s
t
H0 g Ž u . dB Ž u . s
a.s.
Since I y Pt x Ž t q s . is orthogonal to L Ž B, t ., we have E x Ž t q s . B Ž u . 4 s ²Pt x Ž t q s . q w I y Pt x x Ž t q s . , B Ž u . : s ²Pt x Ž t q s . , B Ž u . : sE s
t
½H
0
u
H0
g s Ž ¨ . dB Ž ¨ . ?
t
H0 x
w0, ux
Ž ¨ . dB Ž u .
5
g s Ž ¨ . d¨ .
Also since x Ž t q s . and V Ž t . are independent Žand hence orthogonal., E x Ž t q s . . B Ž u. 4 s
u
H0
E x Ž t q s . .G Ž ¨ , X Ž ¨ . . DŽ ¨ .
d¨
and consequently g s Ž u. s
E x Ž t q s . .G Ž u, X Ž u . . 4 D Ž u.
Ž a.a.u . .
Ž 3.13.
82
S. A. ELSANOUSI
Now, t
x Ž t q s . s Ttu X Ž u . Ž s . q
Hu
TtuDC Ž ¨ . Ž s . dU Ž ¨ . .
Since GŽ u, X Ž u.. is F Ž u. measurable, and F Ž u. is independent of Uq Ž u., we have on using the linearity of Ttu and evaluation at s E x Ž t q s . .G Ž u, X Ž u . . 4 s E Ttu X Ž u . Ž s . .G Ž u, X Ž u . . 4 s Ttu E X Ž u . .G Ž u, X Ž u . . 4 Ž s . s Ttug Ž u . Ž s . . Therefore we have from Ž3.13. XˆŽ t . Ž s . s
t
H0
Ttug Ž u . Ž s . D Ž u.
dB Ž u .
Ž a.s.. .
Remark. The stipulated boundedness conditions on F, G, C and the Holder continuity of c entail that the process Ž s, w . ¬ w Ttug Ž u.xŽ s, v . has ¨ Holder continuous paths. An application of Kolmogorov’s theorem then ¨ ensures the existence of a continuous version of the process Ž s, v . ¬ w H0t ŽTtug Ž u.rDŽ u..Ž s . dBŽ u.xŽ v .. If this version is selected, we will then have the stronger result: XˆŽ t . s
; t g I,
t
H0
Ttug Ž u . D Ž u.
dB Ž u . Ž in C Ž J . . a.s.
Ž 3.14.
Since dBŽ u. s Ž dy Ž u. y GŽ u, XˆŽ u.. du.rDŽ u., we can write Ž3.14. in the form t XˆŽ t . s H 0
Ttug Ž u . D2 Ž u.
dy Ž u . y
t
H0
Ttug Ž u . .G Ž u, XˆŽ u . . D2 Ž u.
du. Ž 3.15.
In order to obtain XˆŽ t . Žand thereby ˆ x Ž t . s XˆŽ t .Ž0.., we need to determine g Ž t . first. By linearity of GŽ t, .., we have
g Ž t . Ž s . s G Ž t , E X Ž t . . X Ž t . Ž s . 4 . y G Ž t , E XˆŽ t . . XˆŽ t . Ž s . 4 . From Theorem 1 X Ž t . Ž s . s Tt0c Ž s . q
t
H0
TtuDC Ž u . Ž s . dU Ž u . .
;s g J.
83
FILTERING FOR A RETARDED EQUATION
Since Tt0c is AŽ0. measurable, and AŽ0. is independent of UqŽ 0., we have ;s g J E X Ž t . Ž s . . X Ž t . Ž s . 4 s E Tt0c . Ž s . . Ž Tt0c . Ž s . 4 t
q
H0
TtuDC Ž u . Ž s . . TtuDC Ž u . Ž s . du.
Since from Theorem 2 XˆŽ t . Ž s . s
Ttug Ž u .
t
H0
Ž s . dB Ž u . ,
D Ž u.
we have E XˆŽ t . Ž s . . XˆŽ t . Ž s . 4 s
t
H0
Ttug Ž u . Ž s . . Ttug Ž u . Ž s . D2 Ž u.
du.
Consequently, we deduce that
g Ž t . Ž s . s G Ž t , E Tt0c . T0t c Ž s . 4 . q G t ,
ž
ž
y G t,
t
H0
Ttug Ž u . . Ttug Ž u . Ž s . D 2 Ž u.
s E G Ž t , Tt0c . Tt0c Ž s . 4 q y
t
H0
ž
G t,
u
u
t
du
t
/
u
u
t
/
Ž s . du
t
H0 G Ž t , T DC Ž u . . T DC Ž u .
Ttug Ž u . . Ttug Ž u . Ž s . D 2 Ž u.
t
H0 T DC Ž u . . T DC Ž u .
t
Ž s . . du
du.
If we define H: I = C Ž J . ª C Ž J . by H Ž t , h . Ž s . s G Ž t , h .h Ž s . . we may write
g Ž t . s E H Ž t , Tt0c . 4 q
t
H0
H Ž t , TtuDC Ž u . . du y
t
H0
ž
H t,
Ttug Ž u . D Ž u.
/
du.
Ž 3.16.
/
84
S. A. ELSANOUSI
By considering the signal x Ž t . y E x Ž t .4 and the observation y Ž t . y H0t GŽ u, E x u4. du, we can relax the requirement EŽ c . s 0 and get the filtering equations XˆŽ t . s E Tt0c 4 q t
y
H0
D 2 Ž u.
Ttug Ž u . D 2 Ž u.
g Ž t . s E H Ž t , Tt0 c y E Ž c . Tt g Ž u .
dy Ž u .
.G Ž u, XˆŽ u . . du
Ž 3.159.
t
. 4 q H H Ž t , TtuDC Ž u . . du 0
u
t
y
Ttug Ž u .
t
H0
H0 H
ž
t,
D Ž u.
/
du.
Ž 3.169.
Finally, to get these equations in the multi-dimensional case, we begin by choosing a symmetric matrix K Ž t . such that K Ž t . 2 s w DŽ t . DŽ t .*xy1 for each t g I. We then set BŽ t. s
t
t
H0 G Ž u, X Ž u . . du q H0 K Ž u . D Ž u . dV Ž u .
g Ž t . s E X Ž t . .G Ž t , X Ž t . . * 4 y E XˆŽ t . .G Ž t , XˆŽ t . . * 4
Ž 3.17. Ž 3.18.
and modify the derivation of Ž3.159., Ž3.169. with no particular difficulties to get THEOREM 3. Gi¨ en the assumptions layed out in Section 2, the solution of the filtering equations Ž3.1. ] Ž3.4. is ˆ x Ž t . s XˆŽ t .Ž0. where XˆŽ t . s E Ž Tt0c . q t
t
H0 T g Ž u . .
H0 T g Ž u . .
y
0
t
u
t
D Ž u. D Ž u. * y1
D Ž u. D Ž u. *
y1
dy Ž u .
G Ž u, XˆŽ u . . du
Ž 3.19.
and
g Ž t . s E H Ž t , Tt0 c y E Ž c . t
t
. 4 q H H Ž t , TtuDC Ž u . . du 0
H0 H Ž t , T g Ž u . . K Ž u . . du.
y
u
t
Ž 3.20.
85
FILTERING FOR A RETARDED EQUATION
Here we ha¨ e H: I = C Ž J, R n . ª C Ž J, LŽR m , R n .. gi¨ en by H Ž t , h . Ž s . s G Ž t , h .h Ž s . * . *.
Ž 3.21.
Remarks. Ž1. Equation Ž3.20. is a non-linear Volterra integral equation with phase space C Ž J, R n .. One suspects that iterative methods for its solution can be obtained along the lines of those for equations with a finite dimensional phase space. Equation Ž3.19. is a stochastic Volterra type integral equation in C Ž J, R n .. If we imbed C Ž J, R n . in L2 Ž J, R n ., then equations of this type afford solutions via the Picard iteration procedure Žcf. w3x.. Ž2. If GŽ t, h . s GŽ t . g Žh . for some m = m matrix GŽ t . and a bounded linear g: C Ž J, R n . ª R m we can write
g Ž t . s b Ž t . G* Ž t . , where b Ž t . s E X Ž t .. g Ž X Ž t ..*4 y E XˆŽ t .. g Ž XˆŽ t ..*4 . An obvious modification of the above argument leads to the equations XˆŽ t . s E Tt0c 4 q t
t
H0 T b Ž u . .G* Ž u . . u
t
H0 T b Ž u . .G* Ž u . .
y
u
t
D Ž u. D Ž u. *
b Ž t . s E h Ž Tt0 c y E Ž c . t
y1
D Ž u. D Ž u. * y1
dy Ž u .
G Ž u . g Ž XˆŽ u . . du Ž 3.22.
t
. 4 q H h Ž TtuDC Ž u . . du 0
H0 h Ž T b Ž u . .G* Ž u . K Ž u . . du,
y
u
Ž 3.23.
t
where hŽh .Ž s . s w g Žh.h *Ž s ..x* ;h g C Ž J, R n ., s g J. In the special case when g s e0 , the evaluation at 0, we get t
ˆx Ž t . s E Ž Tt0c . Ž 0 . 4 q H Ttub Ž u . Ž 0. .G* Ž u . . D Ž u . D Ž u . *
y1
dy Ž u .
0
t
H0 Ž T b . Ž u . Ž 0. .G* Ž u .
y
u
t
D Ž u. D Ž u. *
y1
GŽ u. ˆ x Ž u . du Ž 3.24.
which is a stochastic integral equation with configuration space L 2 Ž V, R n .; and so we would not need to solve for XˆŽ t . in L 2 Ž V, C Ž J, R n .. to obtain the solution ˆ x Ž t . of the filtering problem. The presence of XˆŽ u. in the right hand side of Ž3.19. and Ž3.22. precludes such possibilities even in the
86
S. A. ELSANOUSI
case of fixed delay: g Žh . s h Žyr .. Observe that XˆŽ u.Žyr . s E x Ž u y r . N Y Ž u.4 may not equal ˆ x Ž u y r . s E x Ž u y r . N Y Ž u y r .4 . Ž3. If we set r s 0, then C Ž J, R n . s R n, F Ž t, h . s F Ž t ..h , GŽ t, h . s GŽ t ..h for some bounded matrices F Ž t ., GŽ t ., DC Ž t . s C Ž t ., X Ž t . s x Ž t ., and Ž r t .Ttuh s F Ž t .ŽTtuh .. Using the formula for differentiation under the stochastic integral Žcf. w11, Lemma VI, 4.4x., we recapture the Kalman]Bucy equations viz. dxˆŽ t . s F Ž t . y b Ž t . G* Ž t . D Ž t . D Ž t . * q b Ž t . G* Ž t . D Ž t . D Ž t . * db dt
y1
y1
GŽ t . 4 ˆ x Ž t . dt
dy Ž t .
s FŽ t. b Ž t. q b Ž t. FŽ t.* q CŽ t. CŽ t.* y b Ž t . GŽ t . * DŽ t . DŽ t . *
y1
b *Ž t . GŽ t . .
Ž4. The imposition of boundedness Žrather than square integrability . on the coefficients C, D, F, G and of Holder continuity on the paths of c ¨ Žrather than mere continuity. is motivated by our desire for the process Ž s, v . ¬ w H0t Žw Ttug Ž u.xrDŽ u..xŽ s . dBŽ u.xŽ v . to possess a continuous version so that Ž3.14. would hold in C Ž J .. We do not know in general if continuity of f Ž u, ? . ; u would guarantee the continuity of a version of Ž s, v . ¬ H0t f Ž u, x . dBŽ u.. ACKNOWLEDGMENT The author thanks S. E. A. Mohammed of Southern Illinois University for suggesting the problem in the first place and for some very valuable discussions.
REFERENCES 1. C. T. H. Baker, ‘‘The Numerical Treatment of Integral Equations,’’ Clarendon Press, Oxford, 1977. 2. A. T. Bharucha-Reid, ‘‘Random Integral Equations,’’ Academic Press, New YorkrLondon, 1972. 3. P. L. Chow, Stochastic partial differential equations in turbulence related problems, in ‘‘Probabilistic Analysis and Related Topics,’’ Vol. I, Academic Press, New YorkrSan FranciscorLondon, 1978. 4. J. Elliot, Stochastic calculus and applications, in ‘‘Applications of Mathematics,’’ Vol. 18, Springer-Verlag, New YorkrHeidelbergrBerlin, 1982. 5. J. Hale, Theory of functional differential equations, in ‘‘Applied Mathematical Sciences,’’ Vol. 3, Springer-Verlag, New YorkrHeidelbergrBerlin, 1977.
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6. G. Kallianpur, Stochastic filtering theory, in ‘‘Applications of Mathematics,’’ Vol. 13, Springer-Verlag, New YorkrHeidelbergrBerlin, 1980. 7. V. B. Kolmanovskii and L. E. Shaikhet, State estimate of hereditary stochastic systems, Stochastic Anal. Appl. 7, No. 4 Ž1989., 387]411. 8. R. S. Liptser and A. N. Shirayev, Statistics of random processes, I, in ‘‘Applications of Mathematics,’’ Vol. 5, Springer-Verlag, New YorkrHeidelbergrBerlin, 1982. 9. M. Metivier and J. Pellaumail, ‘‘Stochastic Integration,’’ Academic Press, LondonrNew York, 1980. 10. S. E. A. Mohammed, Stochastic functional differential equations, in ‘‘Pitman Research Notes in Mathematics,’’ Vol. 99, Pitman, BostonrLondonrMelbourne, 1984. 11. B. Oksendal, ‘‘Stochastic Differential Equations: An Introduction with Applications,’’ Springer-Verlag, BerlinrHeidelbergrNew YorkrTokyo, 1985.