TA3 = 1O:OO
FILTERING FOR PIECEWISE LINEAR DRIFT AND OBSERVATION
V&Im E. Bent;
Bell Laboratories
Murray Hill, N.J. 07974
+
Ioannis Karafzas
Department of Mathematical Statistics,
Columbia University
New York, NY 10027
valuesandone-sidedderivatives
of the desireddensityatthe
corners, 3n quantitiesin all. Determination of these weights rests
on continuityacrossthecorners,and
on the “elastic”boundary
conditions, which relate the jump in the gradient at a corner to the
value there. A “jump relation” for potentials is instrumental here;
by itsmeanstheboundaryconditionsandtherepresentationare
transformed into a system
of3n linearVolterraequationsforthe
weights. A simple representation leads to 2n similar equations with
the added virtue that they are of the second kind, so that a classical
theory is available. The nonlinear filtering problem is thus reduced
to solving these equations.
ABSTRACT
The filtering problem for piecewise linear drift and observation
functions is reducedtoaninitial-boundaryvalueproblem.The
“corners”giveriseto
local timeterms.
A finite
numberof
sufficient statistics appear, in the form of the values and one-sided
derivatives of theconditionaldensityatthe“corners”,ormore
generally
in
the
form
of weights
in
representation
a
of the
conditional density by potentials. Both kinds of statistics propagate
accordingtolinearVolterraequations,andmust
be consideredas
infinite-dimensional.
The
theory
developed
here
for
piecewise linear
dynamics
enhances the study of the general nonlinear filtering problem in a
natural way: Nonlinear
functions
can
be approximated
over
boundedintervals
by polygons, toanydegree
of accuracy; by
constructing or calculating the optimal
filter for the approximating
piecewise lineardynamicsasindicated
in thispaper,
one can
conceivably
obtain
very
good
sub-optimal
filters for
general
nonlinear dynamics. That the results extend to many dimensions
is
far from clear, but likely whenever the necessary local times can be
defined.
A simple example is discussed in detail
methods and the structure of the solution.
2. THEFILTER
Weconsiderthenonlinearfilteringproblem
of characterizing
theconditionaldistribution
of E, given the observationrecord
u(yu : 0 5 u 5 r ) under the model
(1)
d5, = f ( S , ) d+ dw, ;
(2)
dy, = h ( E r ) d r + d b ,
where w. and b. areindependent
I . INTRODUCTION
Eo
I! ( w . ,b. 1
; yo = 0
Brownian motionprocessesand
f(.),h ( . ) are continuous, piecewise linear functions
Forseveralyears,research
in stochasticfilteringtheory
has
concentrated on discerningthosecasesfor
which the conditional
density is propagated by afinitenumber of sufficientstatistics, so
that the optimal filter can be implemented by a finite dimensional
dynamicalsystem.Sinceit
is now understoodthatsuchcasesare
“exceptional”, it is natural to inquire what methods can
be used to
analyze the structure of filters in an infinite-dimensional setting.
n
(3)
f’(z)
2 ailA,(z)
i=O
n
, h ’ ( z ) = 2 hil.+(z)
=
iO
for a number n of “elbow” or “corner” points
x I < x 2 < ... < x ,
on the real line ( x o = -co,
= +w), and Ai = ( x i , x i + ] ) .
It is known (c.f. for instance [31) that if F(.)is the distribution
function of t o ,theconditionaldistribution we areseeking has an
(unnormalized) density given by
Thispaperadvancessuchastudy
by consideringthecase
of
piecewise-linear
drift
and
observation
functions
in the model.
Roughly
speaking,
the
infinite-dimensional
character
of this
problem is aresult
of thepresence
of afinitenumber
n of
“corners”, which now actas elasric boundaries fortheconditional
density process. Solution for
the latter in each interval of linearity
(between two such corners), through either the Kallianpur-Striebel
formula or the Zakai equation, leads to a linear parabolic PDE with
lineardriftandquadraticpotential,bothtime-dependent
via the
observations.
The
fundamental
kernel
each
ininterval
is
expressible in terms of a pair of statistics similar to those appearing
in the Kalman filter: a covariance-like r (.) independent of the data,
and
mean-like
a
m(,) driven by them.
The
unnormalized
conditional density can then be written by Green’s theorem in each
interval in terms of its kernel by means of single- and double-layer
potentials. The weightsinthesepotentialsarethe
still unknown
iResearchsupported
to further illustrate the
p(t,z) =
J
q(r,z;o,x)dF(x)
R
interms of the“fundamentalunnormalizeddensity”
expressible by the Kallianpur-Striebel formula as
q(t,z;O,x),
..
by the NationalScienceFoundationunder
NSF MCS-81-03435.
583
0191-2216/81/oooO-0583 $00.75
0 1981 IEEE
Here E W denotesexpectation with respect tothe w. process. In
order to eliminate the stochastic integration against the
y. process
- andthus
bring theobservationpathparametricallyintoour
expressions - we have to integrate by parts. Itas rulecannotbe
h (.) is only piecewise
applied
directly,
however,
because
continuously differentiable. As in Mc Kean 141 the corners of h (.)
give local timecontributions,and
we obtain the "generalized
Tanaka formula"
Initial-boundary value problem:
Toconstructacontinuousfunction
u ( r , z ) = u ( t , z ; O , x ) on
( O , T x R ] , which is C'*2in each of the intervals A,; i = 0,1, ...,n ,
and satisfies
(a)
the
forward
equation
+ (-21 hi?,*
1
2
- - h2(z) -y,f (z)h,)u
in ( 0 , T l ~ 4i ; = 0,1, ...,n ,
and the "integration by parts formula"
(b)theboundaryconditions
(c)the
(9),and
initial condition: lim u ( t , z ; O , x )= 6 ( z - x ) .
'IO
An alternative way toamveattheabove
characterization of
u ( t , z ; O , x ) proceeds via the Zakai equationfor q ( t , r ; O , x ) Dl. In
each of the 4 , ' s q satisfies the stochastic equation
where P;+",(t) is the local timespent
z = x i up to time t .
by theprocess
Substitution of (6) into the K.4. formula (4) and some simple
algebra provide the fundamental density in the form
(7)
4
x + w at
q ( t , z ; o , x ) = eho)Y'u(t,z;O,x),
1
=(iq22-f(~)q2-aiq)dt +h(z)qdyr
9
along with the initial condition lim q ( t , z ; O , x ) = 6 ( z - x ) .
'IO
The
(7) trades
the
(stochastic)
Zakai
"Rozovsky
transformation"
equations for the nonstochastic ones in (10). as one can verify after
bit
a
of stochastic
calculus.
Continuity
of q ( t , . , ; O , x ) and
q 2 ( t , . ; 0 , x )translate into continuity of u ( t , . ; O , x )and the boundary
conditions (9). respectively.
3. A JUMP RELATION FOR HEAT-LIKEPOTENTIALS
In this
section
we construct
the
fundamental
solution
for
parabolic equations of the form (10) and examine the behavior
certain potentials based on it.
To simplify derivationandnotation,
equation
(11)
1
v I =- vZ
2
we considerthe parabolic
+ ( e [ t ) - u z ) v 2 - (-21 h 2 z 2 + g ( t ) z ) v
with a , h real constantsand e(.), g(.) continuousfunctions
t 1 0, and seek its fundamental solution i n the form
(12)
I*
1 '
where
2 &dw, - @
:
d
c is the Girsanovexponent.
2 0
The stochasticrepresentation (8) helps us read off the partial
differential
equations
(in
the
intervals
A i ) and
the
boundary
points x , ) satisfied by thefunction u ( . , . ; O , x ) .
conditions(atthe
The keys are the Feynman-Kac formula and the theory of the socalled elastic Brownian Motion [ 6 , p. 1611. Theformerdiscernsa
drift term
<A(+.)
1
"
is thefundamentalGaussiankernel.Substitution
of theproposed
form of r into (1 1) and annihilation of the coefficients of powers of
z up to order 2 yieids the equations
(13)
rn(S,S,X) = x
a log A
at
from the killing rate. The latter imposes an "elastic condition"
- U A t J i - )+) y , ( h i - h i - l ) u ( t , x i )
at each "corner" point x i , i
=
+ 2ar(t) - h 2 r 2 ( t ) ; r ( 0 ) = 0 (Riccati)
am
+ (h2.r(f-s)-a)m + (e(t)+r(t-s)g(t)j = 0 ,
at
i ( i )= 1
,
n
'
(UAt,Xi+)
r ( t , z ; s , x ) =A(t,s,x). c(r(t-8) , z - m ( t , s , x ) )
~ ( x , t=) ( z ~ t ) - "exp(-x2/2J)
1 ~ i:4 hi21A,(z)-7 h 2 ( z-)Y t f ( zh) i l Aii- (0 Z )
(9)
on
with A ( s , s , x ) = 1, r{O) = O and m ( s , s , x ) = x where
,
from the argument of the Girsanov functional, and a potential term
1
of
+ -21 h 2 ( m 2 + r ( t - s ) ) - a f r n g ( t ) = O
A(s,s,x) = 1 ,
=o
and therefore, with
1,...,n .
Therefore the Filtering problem reduces to the following:
584
o(t)
& exp
;
A (f
,s ,x) = exp
[:
- - h 2 J, (rn2(0,s.x)
+r(e-s+g(e)m(e,s,x)
+a (1-3)
I ,I , ,
and on the similarexpression
I 2 with p ( f ) = 1. Since
i ( 0 ) = 1, we have r d Ir ( r ) 5 r l f , for f sufficiently small, with
In the special case u = 0, g ( f ) = e ( r ) pa 0, we have
1
X
r ( t ) = - tanh(hr) , m ( f , s , x ) =
cosh
h
(f-s)
r o positive. Letting C denote a constant (not necessarily the same
throughoutthepaper),
we have 1121 5 Cq", by virtue of the
Lipschitz continuity of s(,). On the other hand, we decompose I I
as J I J 2 , with
+
1
A(f,s,x)=(cosh h(t-s))-H exp(--hx2tanh h(f-s))
2
Thus
(16)
=
r(t,z;s,x)
[F
{
$
sinh h ( f - s ) ) - f i exp - -
(z2+x2)cosh h(t-s)-2u
sinh h ( t - s )
To estimate J 2 first, consider the ratio
recovering a classical result of Szybiak 151.
"jump
a relation"
for
single-layer
Let us now establish
potentials of the form
r(t,z) = ~o'rft,z+~(f);u,~(u))p(u)du
,
C(z+q), so by choosing z , small
~
we
It can be checkedthat I[l
canguaranteethat
5 1, lexp[-ll 5 e l t l . On theotherhand,
for z small enough, it is easy to establish a bound of the form:
by using the fact that xe-',
x"e-x, e-x are bounded functions on
R+.
To estimate J I , we write it in the form J , ,
' s 1)-s u F(t,z+s(t);u,s(f))du
1 1 2 = Jt-n
r ( I -u )
where
p(t)=
1-
+ J12,with
(lJ12/
5 CT')
,
U(Z),
r(r)
r(f,z) is decomposed as rl(?,z)
+ r2(r,2),with
after the change of variable x
=
d!d (with a
well-defined inverse
Z2
u(x;z) = r-I(xz2), provided q is sufficiently small) and where
585
+ [h(xi)-hrxrl2 + 2 V ( x i ) - ~ i x i I h i y t - hi$?
m(r,t-u(x;I),s(r))
22
By 1'H;pital's rule we verify that l i m f ( x ; t , z )
11.
of (1Oi) in the form
( 2 0 )K ( ' ) ( t , z ; s , x )
1; then dominated
--
som
x-3/2e
10
> 0 sufficiently small:
Let us fix an interval Ai = (xi,xi+Jand use Green's theorem to
find a representation there for the solution u. With
210
Returning to the quantity L , we have 11521
IC v Hand
=0
backward
210
We shall also need a jump relation for the analog
so r,(r,z+s(r);u,s(u)p(u)du
1
Integrating dx over Ai
obtains
of the classical double-layer potential.
p(.)
a K ( i ) + L i K ( ' )= 0,
as
K(') the
andfor
with Li the formal adjoint of L.
corresponding to a drift yi and a potential 6 . We find in the usual
way that on Ai
a1 K ( ' ) ( t , Z ; s , x ) u ( s , x )= 1
( K ( i ) u x ) x- - (K,C')u), - ( y i K % ) ,
as
2
2
which establishes (18).
(19) Proposition: For
= L.u,
at
we havetheforwardequation
0 sufficiently small; therefore,
lim ( L - p ( r ) )
I
ki(u)du
l :
4. REPRESENTATION
l i m 11-11 5 CvH .
11 >
-1j
where k i ( u ) = coefficient of z o in V i ( u , z ) . Jump relationsfor
potentials
against
the new kernel K(') follow readily from
Propositions (17) and (19).
=6
take care of lim J l l = 1. Therefore, for v
for all
I
= r(')(t,z;s,x)exp
2 10
convergence and the integral
2
=
+
The identification h = h i , u = u i , e ( t ) = hiyt - f ( x i )
uixi,
g(t) = hi [h (xi)-hixi+uiyf1 in (11) leads to a fundamental solution
=
(xi,xi+l)and ds from
e
to
f
- c,
one
4+1
continuous and s (.) Lipschitz
J
[K(')(t,s;t--c,x)u(t-c,x)
-~(~)(t,z;-c,x)u(c,x)ldr
4
+ u(r-s)
[z-m ( t ,s ,x)]
r ( t -s)
K,(r,z;s,x) = - K ( t , z ; s , x )
For zeAi let e + 0 togetarepresentation
of u interms of its
values and one-sided derivatives at the boundary points xi and xi+,:
1
(21)
u = u(t,z;O,x)
z-m (I,s,x)
r ( r -s)
We can rewrite the left hand side
of (19) in terms of -K, and
K . O ( r - s ) ; the
latter
will
additional
integrands
of the
form
contribute O ( b ) in the critical range ( r - - b , r ) ; by (17) the -K, term
will give a jump k p ( t ) .
Adapting the results of this section to our problem, we see that
we are to solve in R+xAi the equation
This
representation
yields integral
equations
for
the
unknown
values and one-sided derivatives of u at the boundary points upon
z
xi from above
application of the jump relation (19). First let
to get
-
with the boundary conditions (9) and the initial condition
lim u ( t ,z;O,x) = b ( z - x ) ,
f
ri(t,z)
= hiyf
K ( t , z ) = h?z2
4
-f(xi)
- ai(z--xi)
+ 2hi(h(xi)-hix,+uiyf)z - a i
586
and then extend evenly on z < 0. Equivalently, one can work with
the Rozovskytransformation: $(r ,z) = p ( t ,z).exp(-zy,); z 2 0
and do the problem
1
(25) $dt,z) = 7 $ 2 2 ( t ~ z )
1
+ Y ~ $ ~ ( ~+, 7
z ) CV?
- z 2 ) $ ( t , z ) : t > 0,z > o
(22)
$2(t,0)
+ n$(t,0) = 0
;t
$(OJ)
= P (z 1
;z L O .
It is readily seen that the fundamental solution
equation is
>o
of the transformed
where r(t ,z;s , x ) is the function in (16)
with h = 1, and that it
satisfies the jump relation(Proposition(19)).IntegratingGreen's
identify
a [K$Z(S,X) - K * $ ( s , x ) + 2 Y s K $ ( s , x ) 1 - a ( 2 K I L ( s , x ) )= 0
ax
as
over (o,r)xR+, subjecttothe
initial andboundaryconditions:
$(O,x) = P ( x ) . W , O )
4 ( 0 , $2(t,O) = r 1 4 ( t ) , we get
the
integral representation
I
The elastic condition at xi is
u 2 ( t . x i + 0 )- u2(t,xi+)
(9)
(27) 2$(t,z)
=
JoiK,(t,z;s,O)-y,K(t,r;s,O))4(s)ds+
+yf(hi-hi-l)u(t,xi) = O ,
and we have 3n equations for 3n quantities u 2 ( t ,xi +O),
i = 1,...,n .
u ( f .xi),
Passing to the limit as z l 0 yields, by virtue of the jump relation,
the Volterra equation
It is easy to show by classical jump relation methods that if one
is given 3n quantities u2(r.xi k O ) , u ( t , x i ) satisfying ( 2 2 ) , ( 2 3 ) ,
and (9) then the representation
( 2 3 ) defines a solution of problem
(IO). Thus we have proved the following
( 2 4 ) StructureTheorem: For the "piecewise linear"dynamics
( 1 ) - ( 3 ) , the fundamental unnormalized conditional density has the
form
+ 2 jOm
K ( r , o ; o , x ) p ( x ) d ,~
of
which is of thesecondkindandthusuniquelysolvablefora
continuousfunction @(.). Conversely,starting with this 4(.) we
can define $ ( t , z ) by ( 2 7 ) andcheckthattheequationandthe
initial condition are satisfied. To show that the boundary condition
is also satisfied, we first let z 1 0 in ( 2 7 ) , t o obtain from the jump
relationandinconjunctionwith
( 2 8 ) that: $(r,o) = 4(t); t > 0.
Secondly, we get a new integral representation of $ ( r . z ) in terms of
p ( . ) , +(.) and I L 2 ( r ; ) , which compared with ( 2 7 ) yields
J o l K ( r , z ; s , ~ ) X ( s ) d s=
W ) = $2(t ,o)
with K(')as in ( 2 0 ) , and u,(s,xikO),
integral equations ( 2 2 ) , ( 2 3 ) ,(9).
u ( s , x i ) solutions of the
O :
t
> 0 ,z > 0 ,
+ yf 4 0 ) .
Wedifferentiatethisexpression
with respect to z andthenlet
1 0,to receive by the jump relation (19):
z
I
X(?) = Jo K,(t,o;s,o)X(s)dr ; t
5. A N EXAMPLE
As simple
a example,
we consider
the
case
j ( z ) = 0,
h ( z ) = Iz I, and suppose that the distribution
of t o has a density
p ( . ) which is anevenfunctionon
R . Takingadvantage of the
symmetryintheproblem,
we areseekingtosolvetheinitialboundary value problem for the a k a i equation
1
2
+ z p ( t ,z)dy, ;
= =- py(t
P2(t,0)
=0
;r >O
=P(z)
;z L O
P(OJ
1
t
0;
t
> 0, inaccordance
with Lemma 7 of
Conclusion: the (unnormalized) conditwnal density forthis
problem is
> 0, z > 0
dlp(t,z)
,z)dt
which implies X ( r )
[ll.
> 0,
587
filtering
For the kernels
under
considerations,
Q ( t , s ) and Pl(t,s) are
boundedfunctions,andtherefore
so are L ( r , s ) and L I ( t J).
Consequently, one can formally differentiate:
= p(t,-z)
;t >O;Z
<o
with +(.) the unique solution of the integral equation (28).
6.
ANOTHERAPPROACH
For n > 1, or asymmetric initial data the system of integral
equations (22), (23) for the
weights in the potentials representing
u ( t , z ) in the variousintervals Ai is "of the first kind" inan
unpleasant way that is not easily made to be "of the second kind"
of
by Abel's transformation. To find analternativeequivalentset
equations more amenable to the classical method, we represent the
solution of equation (10)in R+xAj as a superposition of two single
K(j3 andthe
layerpotentialsagainst
thefundamentalsolution
weightfunctions +,?, +GI placed atthe two end-points of the
interval A ~o; Ii 5 n (bo+(t)= b;+l(r)=~);
(291
u ( ' ) ( t , z ) = Jot K ( ' ) ( r , z ; s , x i ) + ? ( s ) d s
+ f,
I
+
K ( i ) ( r , ~ ; s , x i + l ) b i + l ( ~ ) d r + K ( i ) ( t , z ;.O , x )
The aboveexpressionsatisfiesequation
(10) and the requisite
initial condition. Matching the values of u(')(r ;), uci-I)(r ;) across
xi we obtain the Volterra equation of the first kind:
(30)
so'
~(i)(r,x~;s,x~~,.)++(s)dc
+ Jof
+
K ( ' ~ ( t s i ; s , x i + l ~ b i + l ~ s+) d r
K(')(t.Xi,O,X)
=
= J~~
K(i-l)(r,xi;s,xi-1)6i-l(s)ds
+
soL
K(i-l)(t,xi;s.xi)+;(s)dc
+
+ K(+I'(f Ji;O,x) .
This equation can be reduced to one of the second kind via the
Abel transformation. For convenience,
let
us introduce
the
,~~;s,x~+~),
notation: K?(t J ) = K ( ' ) ( r, q ; s , x i ) , Kj;, ( t ,s) =
K Z l ( f , s )= K ( i - 1 ) ( f , q 2 p + - l ) , KF(t.8) = K(i-l)Jr,xi;s,xi) and
K(t,s)+(s)&.
denotegenericallyany of theintegralsin (30)by
0
Now we Abel-transform, i.e. replace r by T , mulbply by (f-r)-',
integrate T over (0,r) andinterchangetheorder
of s and r
integrations to obtain the expression
sof
L(r,s)+(s)ds
helps us integrate by parts:
588
The system of Volterra equations (31). (32) of the second kind is
then
uniquely
solvable
for
2n continuous
functions
(bi’(t),
1 5 i 5 n ) . Thus we have proved the following
Second Structure Theorem: The
fundamental
unnormalized
conditionaldensityfor
the “piecewise-linear” filter in (1)-(3) is
given by
q(t,z;O,x) = e h “ ) ’ ‘ i lAi(z)K(‘)(f,z;s,xi)b+(s)ds
t
i-0
where the 2n continuous weight functions ( b t ( r ) ; 1 5 i 5 n ) are
obtained as the unique solution to the system of integral equations
(31), (321, and & ( r ) = b;+,(t) = 0.
REFERENCES
111 J. R. CANNON and C. D. HILL: “Existence,uniqueness,
stability and monotone dependence in a Stefan problem for
the heat equation”, J. Math. Mech. I7 (19671, 1-19.
121 A. FRIEDMAN: “Portia1 DigPrerrrial Equatwns of Parabolic
Type”, Prentice Hall, Englewood Cliffs, N. J., 1964.
[31 G. KALLIANF’UR: ‘3ochastic Filfering Theory”, Springer
Verlag, Berlin, 1980.
141 H. P. McKEAN: “A Holdercondition for Brownian local
time”, J. Math. Ky&o Univ. 1 (1962). 196-201.
151 A. SZYBIAK: “On the asymptotic behaviour of the solutions
of theequation
Au - & / a t
c ( x ) u =O”, Bull. Acad.
Polon. Sc.. 52;. math.. ad.. phys. 7 (19591, 183-186.
+
[61
D. WILLIAMS ‘‘Lhflusions, Markov Processes and
Martingales” Vol. 1: Foundatiom, J. Wiley & Sons,
Chichester, 1979.
the
optimal
filtering
of diffusion
I71 M. ZAKAI: “On
processes”, 2. Wahrscheinlichkeitstheorie venv. Cebiere 1 1
(1969), 230-243.
589