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Filtering with a small non-linear term in the signal
Systems & Control Letters 15 (1990) 81-90 North-Holland
81
Filtering with a small non-linear term in the signal Robert J. Elliott
*
Department of Statistics and Applied Probability, Unioersity of Alberta, Edmonton, Alberta, Canada T6G 1X5 Received 29 January 1990 Revised 4 May 1990
Abstract: The filtering problem, when the signal and observation processes are linear and have Ganssian noise, has an explicit finite dimensional solution, the Kalman filter. In a recent paper Picard considered the situation where there is in addition a small non-linear term in the signal process. Using a Girsanov transformation on the signal and using homogeneous chaos expansions he estimated the error when related linearizations are considered. In this paper the non-linearity is approximated by considering stochastic flows for the signal and a Girsanov transformation for the observation. The result can be expressed in terms of Gaussian densities.
Keywords: Kalman filter; nonlinear drift; stochastic flow.
1. I n t r o d u c t i o n F o r simplicity we shall consider real valued signal a n d o b s e r v a t i o n processes; the vector case c a n be treated similarly with more complicated n o t a t i o n a n d calculations. Suppose wt and B,, t > O, are two i n d e p e n d e n t B r o w n i a n m o t i o n s defined o n a p r o b a b i l i t y space (12, F, P ) which as a complete, fight c o n t i n u o u s filtration { Ft } to which w a n d B are adapted, a,, t > 0, is a locally integrable, m e a s u r a b l e f u n c t i o n a n d ht, t >__O, is a f u n c t i o n with a locally integrable derivative. C o n s i d e r first a linear signal e q u a t i o n
x t = x S+
a , x , d u + w t.
(1.1)
It will ocassionally be c o n v e n i e n t to write the solution x t of (1.1) as ~s,t(xs). Suppose ~ ( s , t) is the s o l u t i o n of
dCb(s, t ) = a t ~ ( s , dt
t),
t>s,
qb(s, s ) = l .
(1.2)
T h e n ~ ( s , t) = e x p ( f t a , d u ) a n d
xt~'s.t(xs)=~(s,
t)[gs + f s t ~ ( s , u ) - l dwu].
(1.3)
Suppose the o b s e r v a t i o n process { Yt }, t > O, has the form y, -~
£'
h , f ; o . , ( X o ) d s + B t.
(1.4).
W e shall suppose x 0 is a G a u s s i a n F 0 m e a s u r a b l e r a n d o m variable i n d e p e n d e n t of wt, B,, t > O. * Research partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A-7964, and the U.S. Army Research Office, under contract DAAL03-87-K-0102. 0167-6911/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
82
R.J. Elliott / Filtering with non-linear signal
Write Y, = { Ys: s < t } and let { Yt }, t > 0, be the right continuous complete filtration generated by the observations. F o r 0 < s < u N t,
Y, =Ys +
fs'h~,gSs,,(x,) du + B,.
(1.5)
2t(x,) = E[x, Ix,, Yr] for t > s. Then it is known that 2t(x,) is a G a u s s i a n r a n d o m variable for t > s
Write and
where
e,.,=e[x}jx,, Y,] -(e[xtlXs,
Y,l) 2
satisfies the equation
des,, dt
=
2 2 -h,P,.t + 2atPs. t + 1,
P,., = 0.
(1.7)
That is, 2t(x,) is G a u s s i a n with conditional m e a n 2t(x,) and variance Ps.tWe shall write )2t = E[xt [ Yt]. Then 92t is Gaussian with mean and variance Pt given by 2,= Etx0l +
fota,2s ds + fotPsh,(dy,- h,2, d , ) ,
dP,
dt = -hzep'2 + 2atPt + 1,
(1.8)
Po = E[xg] - (E[xo]) z.
(1.9)
Equations (1.6) and (1.7), or (1.8) and (1.9), are forms of the well-known K a l m a n filter. The innovation processes
fl,(Xs)=y,-fsth,2,(x,)du
(t>_s),
flt=y,-fo'hu2, du
(t>_O)
(1.10)
are { Yt } Brownian motions and generate the same filtration as { Yt }. Notation 1.1. Write ~ ( m , P; d x ) for the Gaussian measure on R with m e a n m and variance P. If g is a Borel measurable function on R, write
r(g, m, P ) = f g ( x ) ~ ( m , P; d x ) . For an integrable process
n,(z) If
{ Z~ }, t > O, FIt(Z) will denote the { Yt }-predictable projection of Z, so
= E [ z , i r,] a.s.
g(t, x) is a function such that [g(t, x ) l < K ( l + l x [ "
)
for some positive K, m, then write
Hi(g) = n , ( g ( t , x,)). Finally, we quote the following from [3]:
(1.11)
R.J. Elliott / Filteringwith non-linearsignal
83
Lemma 1.2. (a) Suppose 0 < s < t. The conditional law of x s given Yt is
Ix(mt,, P/, d x ) where m• =.~t = ~-
yuhu dflu,
(1.12)
( ps~2r t 2.2
/)st = P , -
~ ) J~ 3',n, du,
(1.13)
with 31 the solution of y, = 1 +
£'(a, - p, h2)y, ds,
(1.14)
so 3', = expf(~(a, - P,h 2) ds. (b) Suppose g(t, x) is a Borel function such that g and gx satisfy the growth condition (1.11). Then 11t
(Jo
g(s, Xs) ds
) Jo =
t11s(g) ds +
Jo (Jo II,
" )
gx(U, x,,)--~ du y,h s dr,.
(1.15)
Note from (1.3) that the map x ---}~ . t ( x ) is a diffeomorphism of R to itself and O~,.,(x)/Ox = ¢(s, t). The solution of (1.6) can similarly be expressed
2t(Xs)=~(s,
t)[Xs + fst~(s, u ) - l P , , , , h , ( d y , - h , , 2 u ( x ~ ) )
du].
From (1.6), a~,(x~)/Ox s = 3's,t satisfies the equation
3",,, = 1 +
Zt(
2 a~ - P,,~hu)3',, u du,
(1.16)
SO 3"s,t = exp Zt( a u - Ps.uh 2) du.
(1.17)
The random variables ~t(x,) and ~t are Gaussian, and so have finite moments of all orders.
2. Nonlinear signal When the signal and observation equations are of the above linear form the Kalman filter provides the answer to the filtering problem. We shall now consider the situation when a small nonlinearity is introduced into the signal process, as in Picard [3]. Suppose f(t, x) is a measurable function defined on [0, ~ ) x R which is twice differentiable in x and which satisfies the growth condition
If(t, x)l+lfx(t,
x)l
Ixl).
(2.1)
Let e > 0 be a small positive number, and suppose the signal process is now given by the process
~, = x o +
a~Y, + e f ( s , ~,)) ds + w,.
(2.2)
Recall ~0,t(Xo) is the solution of (1.1) and consider the process defined by
z t = x o + fot~(O, s ) - ' e f ( s ,
~o,s(zs)) ds.
(2.3)
R.J. Elliott / Filtering with non-linearsignal
84
Proposition 2.1. The process ~o,t( z,) is the solution of (2.2). Proof. The result follows from the generalized Ito formula of Bismut [1] or Kunita [2]. However, it is immediate in this case: substituting (2.3) in (1.3), ,0.,(z,) = ~(0, t ) { / 0 + f0t~(0, s ) - ' e f ( s , ~o,,(z,))as+ f j ~ ( 0 , S) -1 dw~.]
(2.4)
and the result follows by differentiation. Remark 2.2. Note that because of the linear growth condition (2.1) satisfied by f, ~, = ~o,t(z,) has finite moments of all orders. If Z: is a process we shall write Z: = O(e k) if
))'-
s_
for every p > 1. Proposition 2.3.
Yt - xt = Do, ' = ~o,,(z,) - ~o,t(Xo) = e~(O, t) fot~(O, s ) - l f ( s , ~o,,(Xo)) ds + O(e2). Proof. From (1.3) and (2.4),
Do.' = ~o.,( zt ) - ~o,,( Xo) = e~(O, t ) fot~b(O, s )- ' f ( s, ~o.s( Zs) ) ds.
(2.5)
However,
f ( s , ~O,s(Zs)) - f ( s ,
~o,s(Xo))
=
fOIL(s, ~O.s(XO) + aDo.s) da .Do. s.
Substituting in (2.5), D o , t = e ~ ( O , t)[fot~(o ,
+ e2~(O, t)
S)-lf(s, ,o.s(/o))ds]
[Jo'
dP(O,s)
fofx(s,~o,s(Xo)+aDo.s)da
• ~b(O, s)(foS~(o, u ) - a f ( u , , 0 , u ( z u ) ) d u ) d s ] and the result follows from (2.1). Notation 2.4. Write A0, , = q~(0, t)fdq~(O, s)-lf(s, xs) ds. Remarks 2.5. To consider the effect of the perturbed signal process .~, = ~o,t(z,) on the observations we introduce the measure fi defined by the following Girsanov transformation: Write A~ --- exp
hsDo. , dB, - 5 jo,. s o: ds
and put (d P / d P) ] F, = A~.
85
R.J. Elliott / Filteringwith non-linearsignal
Then Bt =
Bt - fdh,Do.s ds
Y, = foths~o,,(z,)
is a Brownian motion under P, i.e., under P,
ds + Bt,
(2.6)
so the observations now depend on Yt = (oa(Z,) • That is, under fi we have the signal process 2, and the observation process Yt given by (2.6). We wish to estimate ffS[~0.t(z,) I Y,] and by Bayes' theorem this is equal to E[ At~o,t(zt) [ Yt]- ( E [ A~, I Y,] )-1.
Lemma 2.6. A ~,= 1 +
ef~h,Ao. , dB, + O(e2).
Proof. It is known that t
2 /'t
[ fSl
e
)
A; = 1 + f@Ash,Do. , dB~ = 1 + efoh,Do,, dB,+e Joh, Do.,,[Jo hs=Do.,2As2dB,2_ dB,, '"
foh,Ao.sdB,+O(e2),
=l+e
by Proposition 2.3. Following Proposition 3.3 of Picard [31 we have: Lemma 2.7.
Hi(A) -1
=1-eIl,[foth,Ao.sdB,]
+ O(e2).
Proof. If x > - 1,
I--1 _ 1 +x[ _
I-It(A ) -
1 the result follows using Lemma 2.6.
The main result is the estimate of the next theorem. Theorem 2.8.
Writing 2, =
ffS[x, I Yt]
~o.,(zt), x, = (0.,(x0),
= E[ x, l Y,] + eE[ X,foth,Ao., dB, I Yt] + eE[Ao.,, Y , ] - e E [ x ,
Proof.
by Proposition 2.3 and Lemma 2.7.
lY,]Elfoth,Ao,sdB, lY, l
"1- O(e2).
(2.7)
R.J. Elliott / Filtering with non-linear signal
86
Remarks 2.9. These expectations are now all of functions of the original linear process x, under the original measure P and will, therefore, all be expressible in terms of Gaussian measures. For example, E[xt I Y~] = xt is given by the Kalman filter. The remaining terms in (2.7) will now be discussed. Lemma
2.10.
E[zlo.,I Y,I = @(0, t)[fo'¢'(0, s)-llI.(f(s, Xs))ds
+jot.sljo~,.. x.,..:,..lo,o, s,-,,,.,..,]. Proof.
~Ao.,'l--~[O,0 ,,j;O,0 . , ' , , s .~,'.,',]--O,0 ,,j0'O,0s' ~I,'s x,,'l's by Fubini's theorem. From Lemma 1.2(b) this is
)
cb(O, t) Jo' @(0, s) - H , ( f ( s , xs) ) as+ @(o, t) ~ '@(o, s) - H, lJosfx(U, x,)P,,y 2' du h,v s dfls. Notation 2.11. Write r
O(s) =hZ~2,-£(h's+a,h,),
* l ( r ) = f o 0 ( S ) ~ ( 0 , s) ds, r
x(r, s, t) = ~t(x~)f~(s, x,) + ys,,f(s, x,), 1,emma
E
dP2(r, u, v ) = foh2~(0, s ) H , ( x ( t , u, s)) ds.
2.12.
[/5.o...., .t[- ,.,~I.o.,.,l /o',.s.s~,, .~,,.s -Jo'.s(So,'x,.. x.~..:,d.l.,s~...,,:,,s + jo'O,s,O,o..,{£o~o, u,-'-.I:,., x.,ld.),s u) P,,%. f.(u, x.) du )h,.%,dfl,, Jo' (Jo'~O'~" o,,.,io~o,-, +
-
-
17 0
-
x.)) ) du fo'h:a,(o, s)(fo'a,(o, u) -1 H,,(Xs(X.)f(u, ^
/o' (/: ( , . ( , u . v ) n°
-
,.(.. .. v))e(0.
u)
-1 P.y.- 1 du
)
h~y~ dfl~.
Proof. From (1.10), B~ = fit + fghs( ~ - x,) ds, so
. lJo~,Os,, .]- ~ [jo~,o..,., ~] +~ [fo'h~Ao,s(~s-Xs)
ds I Y,].
(2.8)
R.J. Elliott/ Filteringwithnon-linearsignal
87
Now
E thsAo.,d£lYt =E[Bth,zao.tlE] - E
£(h,Ao.,+a~h,Ao.,+h,f(s, G))
dslY, ,
SO
However, Bt is Y~ measurable, so
E[~,h,Ao., I Yt] = Bth,E[Ao,, I Yt],
(2.9)
which can be expressed recursively using Lemma 2.9. Also, from Lemma 1.2(b),
E[fot~hJ(s, x~)ds. Yt] = fot,~hflI,(f(s, x,))ds+ fotl-I~(foSfx(U,xu)Pdy~' du)fl~y,h2 d~,. (2 .lO)
Consider
E[fotO(s)Ao.sdslYt] =fotO(s)dP(O , S)fo~Cl)(O,u)-lE[f(u, Xu) lYt] Write
Gl(U, t) = E[f(u, x~) I Yt]. Then
duds.
(2.11,
as in Picard [3],
t G,(u, t)=H~(f(u, x~))+ V~y;1 Ll-Io(fx(U, x,,))yvh o dBv,
so substituting in (2.11) we have
E[fotO(s)Ao,,ds ' Yt]--fot0(s)~(0,
s)(Jo~(0,
u)-lHu(f(u, x~))du}
ds
+ jo'O(,)o(o s){;o(o, u) ,,.,:,L'~o(,x(u. x.)),o~ ~ ~u} ~ , Write ~l(r) = so that
fffl](s)~(O, s) ds. Then
the stochastic Fubini theorem [4] can be used on the final integral
:jo,0(,,o(o ,{jo.O(Ou) Jo (So +
no
(,,(t)-¢l(u)),(0,
)
u) -1 e~y~-1fx(U, xu) du ho~odK. (2.12)
Finally, t 2
Now, E[xJ(u, (1.6). Write
= Job'S(°'
xu) [Yt] = E[~,(x~)f(u,
a2(u, s, t) = e [ ~ s ( x ~ ) / ( u ,
s)£,(o, s
u)-'e[x,:(u, x~)Ir, l du as.
x~) I Yt] where .~s(xu) is the Gaussian random variable given by
x~) I Y,],
n.J. Elliott/ Filteringwithnon-linearsignal
88
SO
-lfj II,.(X(t, u, s))%,h,, dfl,,.
G2(u, s, t)= Fl~(~Cs(X~)f(u, x~))+ Pu7~ Therefore, substituting,
E
]
h~ao,,X, ds It, = fo'h~4(O, s)
:os4(0, u)
t 2
,
nu(~s(~)f(u, ~ ) ) du
s
+fohs4(O,s)fo4(O,u)-%r:'f'no(x(t,u,s))r.h. aaoauds. (2.13) Using the stochastic Fubini theorem again [4], on the final integral, we have
E
h,Ao.,xsdslY t =Lth24(O,s)L4(O,u)-'H,(£,(xu)f(u, ~t-
/o' (/? no
*2(t, U, U)
xu))au
)
- ,2(/.1, U, U ) ) 4 ( 0 , u) -1 puy~_ I du 7vh,, d•.
(2.14)
Substituting (2.9)-(2.14) in (2.8) the result follows. Remarks 2.13. The remaining term in (2.7) that must be considered is
f[XtfohsAo.s ansi Yt] E[xtfohsAo.s a~sl
Yt)
E[xtfohsAo.s(Xs-Xs) asI J
[::
=EIxthtfl, Ao.tlYtl-E x t £(hsAo.,+ash,Ao., +h,f(s, x , ) d s l Y ,
1
+E[Xtfo'h:~o.,(2s-X,) ds I Y,]
L
O'
t 2
Yt].
Each of these terms will be discussed individually as in Lemma 2.12 to derive the following result: Lemma 2.14.
Write
, 3 ( r , u, o) = f f h ~ 4 ( 0 , s)/7o((~,(x~) + ~,(x~)r~.t)f(u, x~) + ~,(Xs)X~L(u, x~)) as. -'o
Then E
[fothsAo,~dB~iY,] xt
xs)) 4(0, t)fot4(O, s) -i Fls(xt(xs)f(s, ^ + 4(0, t +
17.
X(t, s, t)4(O, S) -1
ds -1
fo'O(s)4(o, .)jo4(O, ~)-'u.(~,(x~)/(u, x.)) au d~
(2.15)
89
R.J. Elliott / Filtering with non-linear signal
+
So' (Jo rto
l
(~,l(t)-~l(u))×(/,
u,t)~(o,
u) -1 P, Yu-1 du ~lvho dflv
^
- foB, h~H,(x,(x,)f(s, x,)) ds -
n~
t 2 - fohs~(O,
x ( t , s, ¢)P,%-IB, h, ds v~h~ dBo s S) f o ~ ( O ,
no .3(. jo,(Jo~(
-1 ^ u)17u(Xt(X,)Xsf(U,
u, o)-~,3(u
u. v))~(o,
x~)) duds
-,-
1
)
Proof. E[xthtfltAo,, I Y,] = htfltE[x, Ao., I Yt] and E[xtAo,tlYt] = ~ ( 0 , t)fotdp(O, s)-lE[xtf(s, xs) IY,] d s = ~(0, /)fo'~(0, s)-lG2(s, t, t) ds, (2.16) where, as above, G2(s, t, r) = E[~t(xs)f(s, Xs) [Yt]- Therefore, 1
t
G2(s, t, t) = I-ls(~,(x~)y(s, x~)) + Psys- fs II°(x(t' s, t)Yvh v dfl o and substituting in (2.16) and changing the order of integration,
E[X, Ao,, [Yt]
=
Xs)) ds ~(0, t) fot~(0, s) -1 17~(Xt(X~)f(s, ^ + cb(O, t)
So'170(J:x(t, s, t)~(O, s) -' P~y~-a ds ty~hodfl o.
(2.17)
Now
=
fot~h~G2(s, t, t) ds
= fo'flshsl-Is(.~t(xs)f(s, xs)) ds + fotl-Io( foVx(t,s, t)Ps)'~-lfl~h~ ds)2tvho dflv.
(2.18)
Next l
0
Substituting for G2(t, u, t) = E[xtf(u, xu) [ Yt] we have
[xtj?~,,~o~ ~,~t ]~ jo'O,~,o,o ,, jo~o~o,u ~ 1 ~
~, ~x~,,~u, x~,~ a, (2.19)
R.J. Elliott / Filteringwithnon-linearsignal
90 Finally, we must consider
E[X,foth~xsAosds 1 ~. y~] = f o i h ~ ( 0 ' S)fo~(0,s u ) ~ E [ x , x s f ( u , xu) IY,] d u a s .
(2.20)
Writing g(xu) = E[~,(Xs) I x , , Y~] we have
E[x,xsf(u, xu)[Yt] = E [ g ( x u ) f ( u ,
x,)[Yt] =G3(u,
t),
say.
Using the Ito rule,
G3(u, t ) = I I , ( g ( x u ) f ( u , x,)) + P~y~-
Ho(Q(u, s, t))yvh,, dfl,,
where
Q(u, s, t ) = g x ( x , ) f ( u , x,) + g(xu)f~(u, x,) = ((~t(x~,) + .~,(xu)Vs,t)f(u, x,,) + g(xu)fx(U, x,)). Substituting into (2.20), writing r
• 3(r, u, u) = f 0 h 2 ~ ( 0 , s)~v(Q(u, s, t)) ds and interchanging the order of integration we have
]
E xtLhsxsAo,,dslY t =
fo'h (o, +
Jo'
Hv
s
S)fo~(O, u)-117ug(Xu)f(u, xu) d u d s
)
( , 3 ( t , u, v) - th3(u, u, v ) ) ~ ( O , u ) - ' P ~ y , - ' du y,,h,, dfl~. (2.21)
The result follows by substituting (2.16)-(2.21) into (2.15).
3. Conclusion As in Picard [3], the first two coefficients in an expansion of the conditional mean in powers of e have been determined. However, by using stochastic flows these coefficients can be expressed explicitly in terms of G a u s s i a n measures.
References [1] J.M. Bismut, A generalized formula of Ito and some other properties of stochastic flows. Z. Wahrsch. Verw. Geb. 55 (1981) 331-350. [2] H. Kunita, Some extensions of Ito's formula, Sem. de Probabilit~sXV, Lecture Notes in Mathematics No. 850 (Springer-Verlag, Berlin-New York, 1980) 118-141. [3] J. Picard, A filtering problem with a small nonlinear term, Stochastics18 (1986) 313-341. [4] J. Szpirglas, Sur l'equivalence d'equations differentielles stochastiques h valeurs mesures intervenant dans le filtrage markovien non lin6aire, Ann. Inst. HenriPoincar~14 (1978) 33-59.
81
Filtering with a small non-linear term in the signal Robert J. Elliott
*
Department of Statistics and Applied Probability, Unioersity of Alberta, Edmonton, Alberta, Canada T6G 1X5 Received 29 January 1990 Revised 4 May 1990
Abstract: The filtering problem, when the signal and observation processes are linear and have Ganssian noise, has an explicit finite dimensional solution, the Kalman filter. In a recent paper Picard considered the situation where there is in addition a small non-linear term in the signal process. Using a Girsanov transformation on the signal and using homogeneous chaos expansions he estimated the error when related linearizations are considered. In this paper the non-linearity is approximated by considering stochastic flows for the signal and a Girsanov transformation for the observation. The result can be expressed in terms of Gaussian densities.
Keywords: Kalman filter; nonlinear drift; stochastic flow.
1. I n t r o d u c t i o n F o r simplicity we shall consider real valued signal a n d o b s e r v a t i o n processes; the vector case c a n be treated similarly with more complicated n o t a t i o n a n d calculations. Suppose wt and B,, t > O, are two i n d e p e n d e n t B r o w n i a n m o t i o n s defined o n a p r o b a b i l i t y space (12, F, P ) which as a complete, fight c o n t i n u o u s filtration { Ft } to which w a n d B are adapted, a,, t > 0, is a locally integrable, m e a s u r a b l e f u n c t i o n a n d ht, t >__O, is a f u n c t i o n with a locally integrable derivative. C o n s i d e r first a linear signal e q u a t i o n
x t = x S+
a , x , d u + w t.
(1.1)
It will ocassionally be c o n v e n i e n t to write the solution x t of (1.1) as ~s,t(xs). Suppose ~ ( s , t) is the s o l u t i o n of
dCb(s, t ) = a t ~ ( s , dt
t),
t>s,
qb(s, s ) = l .
(1.2)
T h e n ~ ( s , t) = e x p ( f t a , d u ) a n d
xt~'s.t(xs)=~(s,
t)[gs + f s t ~ ( s , u ) - l dwu].
(1.3)
Suppose the o b s e r v a t i o n process { Yt }, t > O, has the form y, -~
£'
h , f ; o . , ( X o ) d s + B t.
(1.4).
W e shall suppose x 0 is a G a u s s i a n F 0 m e a s u r a b l e r a n d o m variable i n d e p e n d e n t of wt, B,, t > O. * Research partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A-7964, and the U.S. Army Research Office, under contract DAAL03-87-K-0102. 0167-6911/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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R.J. Elliott / Filtering with non-linear signal
Write Y, = { Ys: s < t } and let { Yt }, t > 0, be the right continuous complete filtration generated by the observations. F o r 0 < s < u N t,
Y, =Ys +
fs'h~,gSs,,(x,) du + B,.
(1.5)
2t(x,) = E[x, Ix,, Yr] for t > s. Then it is known that 2t(x,) is a G a u s s i a n r a n d o m variable for t > s
Write and
where
e,.,=e[x}jx,, Y,] -(e[xtlXs,
Y,l) 2
satisfies the equation
des,, dt
=
2 2 -h,P,.t + 2atPs. t + 1,
P,., = 0.
(1.7)
That is, 2t(x,) is G a u s s i a n with conditional m e a n 2t(x,) and variance Ps.tWe shall write )2t = E[xt [ Yt]. Then 92t is Gaussian with mean and variance Pt given by 2,= Etx0l +
fota,2s ds + fotPsh,(dy,- h,2, d , ) ,
dP,
dt = -hzep'2 + 2atPt + 1,
(1.8)
Po = E[xg] - (E[xo]) z.
(1.9)
Equations (1.6) and (1.7), or (1.8) and (1.9), are forms of the well-known K a l m a n filter. The innovation processes
fl,(Xs)=y,-fsth,2,(x,)du
(t>_s),
flt=y,-fo'hu2, du
(t>_O)
(1.10)
are { Yt } Brownian motions and generate the same filtration as { Yt }. Notation 1.1. Write ~ ( m , P; d x ) for the Gaussian measure on R with m e a n m and variance P. If g is a Borel measurable function on R, write
r(g, m, P ) = f g ( x ) ~ ( m , P; d x ) . For an integrable process
n,(z) If
{ Z~ }, t > O, FIt(Z) will denote the { Yt }-predictable projection of Z, so
= E [ z , i r,] a.s.
g(t, x) is a function such that [g(t, x ) l < K ( l + l x [ "
)
for some positive K, m, then write
Hi(g) = n , ( g ( t , x,)). Finally, we quote the following from [3]:
(1.11)
R.J. Elliott / Filteringwith non-linearsignal
83
Lemma 1.2. (a) Suppose 0 < s < t. The conditional law of x s given Yt is
Ix(mt,, P/, d x ) where m• =.~t = ~-
yuhu dflu,
(1.12)
( ps~2r t 2.2
/)st = P , -
~ ) J~ 3',n, du,
(1.13)
with 31 the solution of y, = 1 +
£'(a, - p, h2)y, ds,
(1.14)
so 3', = expf(~(a, - P,h 2) ds. (b) Suppose g(t, x) is a Borel function such that g and gx satisfy the growth condition (1.11). Then 11t
(Jo
g(s, Xs) ds
) Jo =
t11s(g) ds +
Jo (Jo II,
" )
gx(U, x,,)--~ du y,h s dr,.
(1.15)
Note from (1.3) that the map x ---}~ . t ( x ) is a diffeomorphism of R to itself and O~,.,(x)/Ox = ¢(s, t). The solution of (1.6) can similarly be expressed
2t(Xs)=~(s,
t)[Xs + fst~(s, u ) - l P , , , , h , ( d y , - h , , 2 u ( x ~ ) )
du].
From (1.6), a~,(x~)/Ox s = 3's,t satisfies the equation
3",,, = 1 +
Zt(
2 a~ - P,,~hu)3',, u du,
(1.16)
SO 3"s,t = exp Zt( a u - Ps.uh 2) du.
(1.17)
The random variables ~t(x,) and ~t are Gaussian, and so have finite moments of all orders.
2. Nonlinear signal When the signal and observation equations are of the above linear form the Kalman filter provides the answer to the filtering problem. We shall now consider the situation when a small nonlinearity is introduced into the signal process, as in Picard [3]. Suppose f(t, x) is a measurable function defined on [0, ~ ) x R which is twice differentiable in x and which satisfies the growth condition
If(t, x)l+lfx(t,
x)l
Ixl).
(2.1)
Let e > 0 be a small positive number, and suppose the signal process is now given by the process
~, = x o +
a~Y, + e f ( s , ~,)) ds + w,.
(2.2)
Recall ~0,t(Xo) is the solution of (1.1) and consider the process defined by
z t = x o + fot~(O, s ) - ' e f ( s ,
~o,s(zs)) ds.
(2.3)
R.J. Elliott / Filtering with non-linearsignal
84
Proposition 2.1. The process ~o,t( z,) is the solution of (2.2). Proof. The result follows from the generalized Ito formula of Bismut [1] or Kunita [2]. However, it is immediate in this case: substituting (2.3) in (1.3), ,0.,(z,) = ~(0, t ) { / 0 + f0t~(0, s ) - ' e f ( s , ~o,,(z,))as+ f j ~ ( 0 , S) -1 dw~.]
(2.4)
and the result follows by differentiation. Remark 2.2. Note that because of the linear growth condition (2.1) satisfied by f, ~, = ~o,t(z,) has finite moments of all orders. If Z: is a process we shall write Z: = O(e k) if
))'-
s_
for every p > 1. Proposition 2.3.
Yt - xt = Do, ' = ~o,,(z,) - ~o,t(Xo) = e~(O, t) fot~(O, s ) - l f ( s , ~o,,(Xo)) ds + O(e2). Proof. From (1.3) and (2.4),
Do.' = ~o.,( zt ) - ~o,,( Xo) = e~(O, t ) fot~b(O, s )- ' f ( s, ~o.s( Zs) ) ds.
(2.5)
However,
f ( s , ~O,s(Zs)) - f ( s ,
~o,s(Xo))
=
fOIL(s, ~O.s(XO) + aDo.s) da .Do. s.
Substituting in (2.5), D o , t = e ~ ( O , t)[fot~(o ,
+ e2~(O, t)
S)-lf(s, ,o.s(/o))ds]
[Jo'
dP(O,s)
fofx(s,~o,s(Xo)+aDo.s)da
• ~b(O, s)(foS~(o, u ) - a f ( u , , 0 , u ( z u ) ) d u ) d s ] and the result follows from (2.1). Notation 2.4. Write A0, , = q~(0, t)fdq~(O, s)-lf(s, xs) ds. Remarks 2.5. To consider the effect of the perturbed signal process .~, = ~o,t(z,) on the observations we introduce the measure fi defined by the following Girsanov transformation: Write A~ --- exp
hsDo. , dB, - 5 jo,. s o: ds
and put (d P / d P) ] F, = A~.
85
R.J. Elliott / Filteringwith non-linearsignal
Then Bt =
Bt - fdh,Do.s ds
Y, = foths~o,,(z,)
is a Brownian motion under P, i.e., under P,
ds + Bt,
(2.6)
so the observations now depend on Yt = (oa(Z,) • That is, under fi we have the signal process 2, and the observation process Yt given by (2.6). We wish to estimate ffS[~0.t(z,) I Y,] and by Bayes' theorem this is equal to E[ At~o,t(zt) [ Yt]- ( E [ A~, I Y,] )-1.
Lemma 2.6. A ~,= 1 +
ef~h,Ao. , dB, + O(e2).
Proof. It is known that t
2 /'t
[ fSl
e
)
A; = 1 + f@Ash,Do. , dB~ = 1 + efoh,Do,, dB,+e Joh, Do.,,[Jo hs=Do.,2As2dB,2_ dB,, '"
foh,Ao.sdB,+O(e2),
=l+e
by Proposition 2.3. Following Proposition 3.3 of Picard [31 we have: Lemma 2.7.
Hi(A) -1
=1-eIl,[foth,Ao.sdB,]
+ O(e2).
Proof. If x > - 1,
I--1 _ 1 +x[ _
I-It(A ) -
1 the result follows using Lemma 2.6.
The main result is the estimate of the next theorem. Theorem 2.8.
Writing 2, =
ffS[x, I Yt]
~o.,(zt), x, = (0.,(x0),
= E[ x, l Y,] + eE[ X,foth,Ao., dB, I Yt] + eE[Ao.,, Y , ] - e E [ x ,
Proof.
by Proposition 2.3 and Lemma 2.7.
lY,]Elfoth,Ao,sdB, lY, l
"1- O(e2).
(2.7)
R.J. Elliott / Filtering with non-linear signal
86
Remarks 2.9. These expectations are now all of functions of the original linear process x, under the original measure P and will, therefore, all be expressible in terms of Gaussian measures. For example, E[xt I Y~] = xt is given by the Kalman filter. The remaining terms in (2.7) will now be discussed. Lemma
2.10.
E[zlo.,I Y,I = @(0, t)[fo'¢'(0, s)-llI.(f(s, Xs))ds
+jot.sljo~,.. x.,..:,..lo,o, s,-,,,.,..,]. Proof.
~Ao.,'l--~[O,0 ,,j;O,0 . , ' , , s .~,'.,',]--O,0 ,,j0'O,0s' ~I,'s x,,'l's by Fubini's theorem. From Lemma 1.2(b) this is
)
cb(O, t) Jo' @(0, s) - H , ( f ( s , xs) ) as+ @(o, t) ~ '@(o, s) - H, lJosfx(U, x,)P,,y 2' du h,v s dfls. Notation 2.11. Write r
O(s) =hZ~2,-£(h's+a,h,),
* l ( r ) = f o 0 ( S ) ~ ( 0 , s) ds, r
x(r, s, t) = ~t(x~)f~(s, x,) + ys,,f(s, x,), 1,emma
E
dP2(r, u, v ) = foh2~(0, s ) H , ( x ( t , u, s)) ds.
2.12.
[/5.o...., .t[- ,.,~I.o.,.,l /o',.s.s~,, .~,,.s -Jo'.s(So,'x,.. x.~..:,d.l.,s~...,,:,,s + jo'O,s,O,o..,{£o~o, u,-'-.I:,., x.,ld.),s u) P,,%. f.(u, x.) du )h,.%,dfl,, Jo' (Jo'~O'~" o,,.,io~o,-, +
-
-
17 0
-
x.)) ) du fo'h:a,(o, s)(fo'a,(o, u) -1 H,,(Xs(X.)f(u, ^
/o' (/: ( , . ( , u . v ) n°
-
,.(.. .. v))e(0.
u)
-1 P.y.- 1 du
)
h~y~ dfl~.
Proof. From (1.10), B~ = fit + fghs( ~ - x,) ds, so
. lJo~,Os,, .]- ~ [jo~,o..,., ~] +~ [fo'h~Ao,s(~s-Xs)
ds I Y,].
(2.8)
R.J. Elliott/ Filteringwithnon-linearsignal
87
Now
E thsAo.,d£lYt =E[Bth,zao.tlE] - E
£(h,Ao.,+a~h,Ao.,+h,f(s, G))
dslY, ,
SO
However, Bt is Y~ measurable, so
E[~,h,Ao., I Yt] = Bth,E[Ao,, I Yt],
(2.9)
which can be expressed recursively using Lemma 2.9. Also, from Lemma 1.2(b),
E[fot~hJ(s, x~)ds. Yt] = fot,~hflI,(f(s, x,))ds+ fotl-I~(foSfx(U,xu)Pdy~' du)fl~y,h2 d~,. (2 .lO)
Consider
E[fotO(s)Ao.sdslYt] =fotO(s)dP(O , S)fo~Cl)(O,u)-lE[f(u, Xu) lYt] Write
Gl(U, t) = E[f(u, x~) I Yt]. Then
duds.
(2.11,
as in Picard [3],
t G,(u, t)=H~(f(u, x~))+ V~y;1 Ll-Io(fx(U, x,,))yvh o dBv,
so substituting in (2.11) we have
E[fotO(s)Ao,,ds ' Yt]--fot0(s)~(0,
s)(Jo~(0,
u)-lHu(f(u, x~))du}
ds
+ jo'O(,)o(o s){;o(o, u) ,,.,:,L'~o(,x(u. x.)),o~ ~ ~u} ~ , Write ~l(r) = so that
fffl](s)~(O, s) ds. Then
the stochastic Fubini theorem [4] can be used on the final integral
:jo,0(,,o(o ,{jo.O(Ou) Jo (So +
no
(,,(t)-¢l(u)),(0,
)
u) -1 e~y~-1fx(U, xu) du ho~odK. (2.12)
Finally, t 2
Now, E[xJ(u, (1.6). Write
= Job'S(°'
xu) [Yt] = E[~,(x~)f(u,
a2(u, s, t) = e [ ~ s ( x ~ ) / ( u ,
s)£,(o, s
u)-'e[x,:(u, x~)Ir, l du as.
x~) I Yt] where .~s(xu) is the Gaussian random variable given by
x~) I Y,],
n.J. Elliott/ Filteringwithnon-linearsignal
88
SO
-lfj II,.(X(t, u, s))%,h,, dfl,,.
G2(u, s, t)= Fl~(~Cs(X~)f(u, x~))+ Pu7~ Therefore, substituting,
E
]
h~ao,,X, ds It, = fo'h~4(O, s)
:os4(0, u)
t 2
,
nu(~s(~)f(u, ~ ) ) du
s
+fohs4(O,s)fo4(O,u)-%r:'f'no(x(t,u,s))r.h. aaoauds. (2.13) Using the stochastic Fubini theorem again [4], on the final integral, we have
E
h,Ao.,xsdslY t =Lth24(O,s)L4(O,u)-'H,(£,(xu)f(u, ~t-
/o' (/? no
*2(t, U, U)
xu))au
)
- ,2(/.1, U, U ) ) 4 ( 0 , u) -1 puy~_ I du 7vh,, d•.
(2.14)
Substituting (2.9)-(2.14) in (2.8) the result follows. Remarks 2.13. The remaining term in (2.7) that must be considered is
f[XtfohsAo.s ansi Yt] E[xtfohsAo.s a~sl
Yt)
E[xtfohsAo.s(Xs-Xs) asI J
[::
=EIxthtfl, Ao.tlYtl-E x t £(hsAo.,+ash,Ao., +h,f(s, x , ) d s l Y ,
1
+E[Xtfo'h:~o.,(2s-X,) ds I Y,]
L
O'
t 2
Yt].
Each of these terms will be discussed individually as in Lemma 2.12 to derive the following result: Lemma 2.14.
Write
, 3 ( r , u, o) = f f h ~ 4 ( 0 , s)/7o((~,(x~) + ~,(x~)r~.t)f(u, x~) + ~,(Xs)X~L(u, x~)) as. -'o
Then E
[fothsAo,~dB~iY,] xt
xs)) 4(0, t)fot4(O, s) -i Fls(xt(xs)f(s, ^ + 4(0, t +
17.
X(t, s, t)4(O, S) -1
ds -1
fo'O(s)4(o, .)jo4(O, ~)-'u.(~,(x~)/(u, x.)) au d~
(2.15)
89
R.J. Elliott / Filtering with non-linear signal
+
So' (Jo rto
l
(~,l(t)-~l(u))×(/,
u,t)~(o,
u) -1 P, Yu-1 du ~lvho dflv
^
- foB, h~H,(x,(x,)f(s, x,)) ds -
n~
t 2 - fohs~(O,
x ( t , s, ¢)P,%-IB, h, ds v~h~ dBo s S) f o ~ ( O ,
no .3(. jo,(Jo~(
-1 ^ u)17u(Xt(X,)Xsf(U,
u, o)-~,3(u
u. v))~(o,
x~)) duds
-,-
1
)
Proof. E[xthtfltAo,, I Y,] = htfltE[x, Ao., I Yt] and E[xtAo,tlYt] = ~ ( 0 , t)fotdp(O, s)-lE[xtf(s, xs) IY,] d s = ~(0, /)fo'~(0, s)-lG2(s, t, t) ds, (2.16) where, as above, G2(s, t, r) = E[~t(xs)f(s, Xs) [Yt]- Therefore, 1
t
G2(s, t, t) = I-ls(~,(x~)y(s, x~)) + Psys- fs II°(x(t' s, t)Yvh v dfl o and substituting in (2.16) and changing the order of integration,
E[X, Ao,, [Yt]
=
Xs)) ds ~(0, t) fot~(0, s) -1 17~(Xt(X~)f(s, ^ + cb(O, t)
So'170(J:x(t, s, t)~(O, s) -' P~y~-a ds ty~hodfl o.
(2.17)
Now
=
fot~h~G2(s, t, t) ds
= fo'flshsl-Is(.~t(xs)f(s, xs)) ds + fotl-Io( foVx(t,s, t)Ps)'~-lfl~h~ ds)2tvho dflv.
(2.18)
Next l
0
Substituting for G2(t, u, t) = E[xtf(u, xu) [ Yt] we have
[xtj?~,,~o~ ~,~t ]~ jo'O,~,o,o ,, jo~o~o,u ~ 1 ~
~, ~x~,,~u, x~,~ a, (2.19)
R.J. Elliott / Filteringwithnon-linearsignal
90 Finally, we must consider
E[X,foth~xsAosds 1 ~. y~] = f o i h ~ ( 0 ' S)fo~(0,s u ) ~ E [ x , x s f ( u , xu) IY,] d u a s .
(2.20)
Writing g(xu) = E[~,(Xs) I x , , Y~] we have
E[x,xsf(u, xu)[Yt] = E [ g ( x u ) f ( u ,
x,)[Yt] =G3(u,
t),
say.
Using the Ito rule,
G3(u, t ) = I I , ( g ( x u ) f ( u , x,)) + P~y~-
Ho(Q(u, s, t))yvh,, dfl,,
where
Q(u, s, t ) = g x ( x , ) f ( u , x,) + g(xu)f~(u, x,) = ((~t(x~,) + .~,(xu)Vs,t)f(u, x,,) + g(xu)fx(U, x,)). Substituting into (2.20), writing r
• 3(r, u, u) = f 0 h 2 ~ ( 0 , s)~v(Q(u, s, t)) ds and interchanging the order of integration we have
]
E xtLhsxsAo,,dslY t =
fo'h (o, +
Jo'
Hv
s
S)fo~(O, u)-117ug(Xu)f(u, xu) d u d s
)
( , 3 ( t , u, v) - th3(u, u, v ) ) ~ ( O , u ) - ' P ~ y , - ' du y,,h,, dfl~. (2.21)
The result follows by substituting (2.16)-(2.21) into (2.15).
3. Conclusion As in Picard [3], the first two coefficients in an expansion of the conditional mean in powers of e have been determined. However, by using stochastic flows these coefficients can be expressed explicitly in terms of G a u s s i a n measures.
References [1] J.M. Bismut, A generalized formula of Ito and some other properties of stochastic flows. Z. Wahrsch. Verw. Geb. 55 (1981) 331-350. [2] H. Kunita, Some extensions of Ito's formula, Sem. de Probabilit~sXV, Lecture Notes in Mathematics No. 850 (Springer-Verlag, Berlin-New York, 1980) 118-141. [3] J. Picard, A filtering problem with a small nonlinear term, Stochastics18 (1986) 313-341. [4] J. Szpirglas, Sur l'equivalence d'equations differentielles stochastiques h valeurs mesures intervenant dans le filtrage markovien non lin6aire, Ann. Inst. HenriPoincar~14 (1978) 33-59.