LIDS-P-1174
January 1982
OPTIMAL CONTROL AND NONLINEAR FILTERING
FOR NONDEGENERATE DIFFUSION PROCESSES
by
Wendell H. Fleming*
Lefschetz Center for Dynamical Systems
Division of Applied Mathematics
Brown University
Providence, Rhode Island 02912
and
Sanjoy K. Mitter**
Department of Electrical Engineering
and Computer Science
and
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
September 9, 1981
*The research of the first author was supported in part by the Air Force Office of
Scientific Research under AF-AFOSR 76-3063D and in part by the National Science
Foundation, MCS-79-03554, and in part by the Department of Energy, DOE/ET-76-A-012295.
**The research of the second author was supported by the Air Force Office of
Scientific Research under Grant AFOSR 77-3281-D.
Submitted to STOCHASTICS.
1. Introduction
We consider an n-dimensional signal process
(Xl(t),---,xn(t))
and a 1-dimensional observation process
x(t) =
y(t),
obeying
the stochastic differential equations
(1.1)
dx = b[x(t)]dt + a[x(t)]dw
(1.2)
dy = h[x(t)]dt + dw, y(O) = 0,
with
n, 1.
w, w independent standard brownian motions of respective dimensions
(The extensions to vector-valued
y(t)
need only minor modifications.)
The Zakai equation for the unnormalized conditional density
dq = A qdt + hqdy,
(1.3)
where
A
example.
q(x,t) is
t > O,
is the generator of the signal process x(t) .
See [3]
for
By formally substituting
q(x,t) = exp [y(t)h(x)]p(x,t)
(1.4)
one gets instead of the stochastic partial differential equation (1.3) a
linear partial differential equation
(1.5)
with
Pt =
tr a(x)p
of the form
x
p(x,O) = p (x) the density of
a(x) = a(x)a(x)'
+
x(O).
,
Vg(x,t)p,
Here
"u
=(P li
P
t > 0,
)
n
tr a(X)Pxx
p
aij(X)Px.x.
'
i,j=l
Explicit formulas for
gY, VY
are given in
1
§6.
Equation (1.5) is the
basic equation of the pathwise theory of nonlinear filtering.
See [2] or
-2-
[9].
The superscript
y = y(.).
y
indicates dependence on the observation trajectory
Of course, the solution
p = pY
also depends on
y
nx n
We shall impose in (l.l)the nondegeneracy condition that the
matrix
a, h, p0
a(x)
has a bounded inverse
will be stated later.
a
(x).
Certain unbounded functions
allowed in the observation equation (1.2).
polynomial in
x = (Xl,---,Xn)
Other assumptions on
For example,
such that
h(x)J + o
as
h
h
b,
are
can be a
Ix
+-o
.
The connection between filtering and control is made by considering the
function
S =--log p. This logarithmic transformation changes (1.5) into
a nonlinear partial differential equation for
below.
S(x,t), of the form (2.2)
We introduce a certain optimal stochastic control problem for
which (2.2) is the dynamic programming equation.
In
§3
upper estimates for
S(x,t) as
ixi +
-
are obtained, by
using an 'easy Verification Theorem and suitably chosen comparison controls.
Note that an upper estimate for
S
A lower estimate for
Jx[ + X
S(x,t) as
gives a lower estimate for
is obtained in
method from a corresponding upper estimate for
p(x,t).
applied to the pathwise nonlinear filter equation in
2. The logarithmic transformation.
§5
p = -log S.
by another
These results are
§6.
Let us consider a linear parabolic
partial differential equation of the form
Pt =
(2.1)
2tra(x)p
+ g(x,t)
+ V(x,t)p,
t > 0,
p(x,O) = p (x).
When
g = gY,
V = VY
this becomes the pathwise filter equation (1.5), to
§6.
which we return in
p E C2 '1 , i.e. with
"classical" solution
i,
j -
PX.
xx'
Pt
continuous,
,n.
l,
If
p(x,t) to (2.1) we mean a
By solution
p
is a positive solution to (2.1), then
S = -log p
satisfies
the nonlinear parabolic equation
(2.2)
St
=-Ly Qx)Sx
+
H(xt,S,)
t
>
0
S(x,O) = S (x) = -log p (x),
H(x,t,S)
Conversely, if
S(x,t)
= g(x,t)- S 1-S'
a(x)Sx - V(x,t).
is a solution to (2.2), then
p = exp(-S)
is a
solution to (2.1).
For example., if
This logarithmic transformation is well known.
g = V = 0,
then it changes the heat equation into Burger's equation [8].
0 < t < tl,
We consider
x
P
[O,tl].
with
tl
7
We say that a function
with domain
is continuous and, for every compact
uniform Lipschitz condition on
K
fixed but arbitrary.
K E Rn ,
for 0 < t < t1.
satisfies a polynomial growth condition of degree
if there exists
M
Throughout this section and
§3
MI(l+lxlr),
if
P(it) satisfies a
4
We say that
r, and write
1 E
r'
, a
-1
all (x,t) E Q.
the following assumptions are made.
Somewhat different assumptions are made in
a
is of class -
such that
P(x,t)I <
~(2.3)
Q
Q = Rn
Let
are bounded,
§ 's 4,5 as needed.
Lipschitz functions on
We assume:
n
R
-4-
For some
m > 1
(2.4)
g E2 n,
_>
For some
m.
nffm
v E
2m
O
(2.5)
c2
SO
n
.
M1 ,
For some
V(x,t) < M
(2.6)
S (x) > -M
We introduce the following stochastic control problem, for which
(2.2) is the dynamic programming equation.
The process
i(t)
being
controlled is n-dimensional and satisfies
dC = u(Q(q),T)dT + a[C(:T)]dw
(2.7)
, 0 < c < t,
i(0) = x.
The control is feedback,
Rn-valued:
u(T) = u((T),T)
.
(2.8)
Thus, the control
any
u
u
is just the drift coefficient in (2.7).
of class snfl
growth of
fu(x,t) I
1
as
fxj
Xo
.
(2.7) has a pathwise unique solution
r > 0. Here
I|
lt
u E.1
Note that
.
implies at most linear
For every admissible
i
We admit
such that
EI
KI I
u , equation
X
is the sup norm on [0,t].
Let
(2.9)
L(xt,u)
-
1 (u-g(x,t))'aa-1 (x)(u-g(x,t)) - V(x,t).
for every
(x,t) E Q
For
(2.10)
and
u
admissible, let
J
J(x,t,u) = E
L[C(T),t-T
u(T)]dT + S
M[I(t)]
The polynomial growth conditions in (2.4), (2.5) imply finiteness of
.
J
u° P minimizing
The stochastic control --problem is to find
u°P
Under the above assumptions, we cannot claim that an admissible
exists minimizing
J(x,t,u).
J(x,t,u). However, we recall from [7,Thm. VI 4.1] the
following result, which is a rather easy consequence of the Ito differential
rule.
Verification Theorem.
C2
1
n
with
Let
S(x,O) = SO(x).
be a solution to (2.2) of class
Then
(a)
S(x,t) < J.(x,t; u)
(b)
If
u0P
J(x,t;
In
S
§3
= g - aSX
*we use (a) to get upper estimates for
we have defined admissibility,
For
[Sxl
one could generalize (b).
will be used in §3
In §4
u° P
S(x,t) =
S(x,t), by choosing
to be admissible, in the sense
can grow at most linearly with
can grow at most quadratically.
admissible controls to include certain
(a)
is admissible, then
u
u°P).
judiciously comparison controls.
hence S(x,t)
for all admissible
u
Ixi
;
By enlarging the class of
with faster growth as IxI +
-o
HIowever, we shall not do so here, since only part
to get an estimate for
S.
we consider the existence of a solution
S
with the polynomial
growth condition required in the Verification Theorem.
As in [6] we call a control problem with dynamics (2.7) a problem of
-6-
stochastic calculus of variations.
of "average" time-derivative of
which would appear in
i(Tl)
u(;(T),T)
The control
is.a kind
i(T), replacing the nonexistent derivative
the corresponding calculus of variations
a = 0.
problem with
Other control problems.
There are other stochastic control problems
for which (2.2) is also the dynamic programming equation.
which is appealing conceptually, is to rcquire
On choice,
instead of (2.7)that C(T)
satisfy
(2.11)
with
d
i(0)
= x.
We then take
L(x,t,u) =
(2.12)
The feedback control
When
in
3.
{g[(T) ,T] + u[.(T),T] J dT + c[C(T)]dw
=
a = identity,
u
1
u'aa
-1
(x)u - V(x,t).
changes the drift in (211) from
Ju-
L =
V(x,t)
S(x,t).
S(x,t)
m > 1, Z > 0
(2.5).
Theorem 3.1
Let
S(x,0) = SO(x).
(i)
(ii)
S
in (2.4),
Then there exist positive
Let
S(x,t) < l
class
M1,
M2
S(x,t) < M 1 (l+ JxlJ)
0 < t0 < t1
V(x,t).
JxJ + o
as
be a solution oof
For (x,t)E Q,
g + u.
In this section we obtain the following
upper estimates for the growth of
constants
to
corresponds to an action integral
classical mechanics with time-dependent potential
Upper estimates for
g
, m > 1.
+l).
2 (l+xm
in terms of the
C '
n r,
with
such that:
p = max(m+l,).
with
For (x,t) E Rn
x
[t0,tl],
-7-
In the hypotheses of this theorem,
polynomial growth as
that
r
can be
Ixj -+
replaced by
S(x,t) is assumed to have
with some degree
r.
P , or indeed by
m+l
Purely formal arguments suggest that
m+l
confirmed by the lower estimate for
S(x,t)
Proof of Theorem 3.1.
The theorem states
provided
t > t0 > 0.
is best possible, and this is
made in
We first consider
§5.
m > 1. By (2.3)-(2.6)
and (2.9),
L(x,t,u) < Bl(l+lx 2m+1u12 )
(3.1)
so (x) < Bl(l+xlt)
for some
B1.
x E R
Given
u(T), O < T < t.
Let
we choose the following open loop control
u(T) = n(T), where the components
ni(T), satisfy
the differential equation
i = -(sgn Xi)lni m i = 1,---,n,
(3.2)
with
(0O)= x.
From (2.7)
i(l) = ln() + C(T) ,
0 < T < t,
f
·
C
=
Since
a
n(
'for each
r.
By explicitly integrating
m > 1, that
2md
<
ft
d i()
O2m<
E (T)I2 , ds_< 22m
0
_
at[(6)]dw(G).
|<
E|[|[r
is bounded,
(3.2) we find, since
)
1
IIm+l
m+
0
[
)2mdT
I T1
E
+ E
<
f
I
1
I m+l
)
m d
<3s·l+]x
-.I ~-·T I ~ds
0~~~~~~~~
t)2
<
-8-
M3
for some
Since
u.
2
2m
2
= n=
i
2
.u(T)I2dT
Since
In(t)j
K
.
m+1
<
Xlmxl
< Ixi,
< E(Ixj
EJ[(t)[!
for some
n
From (2.10.),
M1.
p = max(m+l, ')
By part (a) of the Verification Theorem, S(x,t) < J(x,t,u),
which implies (i) when
t > t>
For
x = p(0).
bounds
)
(3.1) we get
J(x,t,u) < Mi(l+lxlP),
for some
<_ (1 + Ix
l1
+li(t))
Since
m > 1.
0,
i(t) =
EXSO[ (t)]
is bounded by a constant not depending on
n(t)[
(t) + C(t),
and
EXl(t)Il
by a constant not depending on
is bounded, this
x . The estimates above
and part (a) of the Verification Theorem then give (ii).
It remains to prove (i) when
u(T
_EO.
When
quadratically as
m = 1 ,
lxi
-
g
m = 1.
Consider the "trivial" control
grows at most linearly and
Moreover,
.
E
11
2
t
V
K(l+x)
at most
2
for some K
Using again (a) of the Verification Theorem, we get again (i) with
max(2,e).
[When
p =
m = 1, this is a known result, obtained without using
stochastic control arguments.]
4.
An existence theorem.
In this section we give a stochastic control
proof of a theorem asserting that the dynamic programming equation (2.2) with
-9-
S O. has a solution
the initial data
S
.
taken from [4, p. 222 and top p. 223.]
The argument is essentially
Since (2.2) is equivalent to
0
p ,
the linear equation (2.1), with positive initial data
existence of
to (2.1),
S
from other results which give existence of positive solutions
see [10][11].
However, the stochastic control proof gives a
polynomial growth condition on
Let
Ca
0 < a < 1.
r
(x',t') E
We say that
$
used in the Verification Theorem (§2).
$
r
For any compact
with domain
c Q, there exists
-- (x,t)l < M[It'-tlI/2 +
is of class
2,1
C~'
if
c,
i
X.
, i, j=l,
,
such
are of
of class
class
are
3
n.
In this section the following assumptions are made.
is assumed constant.
By a change of variables in
The matrix
Rn
we may take
C
on
a(x)
= identity
(4-.2)
For fixed
g
M
x'?-xi ].
x
x.x.'St
1
Ca
.Q is of class
imply
.P(x',t')
(4.1)
S
We say that a function
if the following holds.
that (x,t),
one could get
gXi
'
V, Vxi,
,
t
i
g(.,t), V(-,t) are of class
1=
l,--,n, are of class
Ca
Rn
and
c E (0,1].
for some
Moreover,
Ig(x,t)l
(4.3)
with
Y2
2
1x
m
,
small enough that (4.8) below holds. (If
then we can take
(4.4)
< ¥1 +
Y2
m
alxl
m
>
1,
g ECJ~
arbitrarily small.)---- Weassume that
_ a2 < -V(x,t) < A(
+ Ix
2m )
with
P < m,
-10-
for some positive
al, a2 , A
(4.5)
and that
gx E
0
3
S E C
We assume that
n A
im
(4.7)
IS
for some positive
Example.
Vx E
2m
> 0, and
for some
.
(4.6)
m'
S (x) = + o
0
< CS
0
+
C2
C1, C 2
V(x,t) = -kV (x) + V1 (x,t) with
Suppose that
2m, k > 0, and
positive, homogeneous polynomial of degree
polynomial in
functions of
< m-l in
x
t
V (x) a
Vl(x,t) a
of degree <2m-1
with coefficients I61lder continuous
.
g(x,t)
Suppose that
is a polynomial of degree
x , with coefficients if6lder continuous in
is a polynomial of degree
£
satisfying (4.6).
t
, and
S (x)
Then all of the above
assumptions hold.
From (2.9),
L =
(4.2),
lu-g
- V .
If Y2
in (4.3) is small
enough, then
(4.8)
1 (lul
for suitable positive
2+xl
81,
2 )
B2,
Igxl
formation on
B.
- Vg
1
I 2
< 2 lul
2
denotes the operator norm of
Rn .
<
B(1
lul2 +x12m)
Mdreover,
Lx = -g (u-g)
ILx
where
2 < L(x,t,u)
gx
From (4.3), (4.5), (4.8)
+
1
2
2g + IVx
regarded as a linear trans-
ILx
(4.9)
for some positive
Theorem 4.1
C1 ,C2
such that
S(x,t) e
Proof.
constraint
(which we may take the same as in (4.7).)
r = max (2m,Z).
Let
S(x,O) = S0 (x)
data
< ClL + C2
has a unique solution
as jxj_-
<
k
S(x,t)
uniformly for
We follow [4, §5].
Mu.
Then equation (2.2) with initial
For
of class
0 < t < t
C2,1r
' n
.
k = 1,2,---, let us impose the
on the feedback controls admitted-as drifts in (2.7 ).
L~t
Sk(x,t) = min
(4.10)
J(x,t; u)
kul<k
Then
Sk
is a
2,1
C21 solution to the corresponding dynamic;.programming
.equation
(4.11)
(Sk)
t
=
A Sk + Hk(x,t,(Sk) ),
Hk(x,t,(Sk)x) =
min
[L(x,t,u)+(Sk)xu]].
ju_<k
The initial data are again
Sk(x,O) = S (x).
The minimum in (4.10) is
attained by an admissible
ukP.
172].
Now
S1 > S 2 > --- ;
and
bounded below by (4.6), (4.9).
is bounded uniformly
Sk
See [7, p.
is bounded below since
Let
S = lim
k-K
L
and
Sk. Let us show that
for (x,t) in any compact set.
SO
are
(S)x
Once this is established
standard argunients in the theory of parabolic partial differential equations
imply that
2,1
S -E C21
and
S
probabilistic representatior
satisfies (2.2).
For
(Sk)x
there is the
-12-
(Sk)x(x,t) = Ext
(4.12)
where
5k
is the'solution to (2.7) with
uk ()
k
u =
differentiating (4.10) with respect to
Another proof, based on
ukP
+ C (t+l)[
L[ k(T),t-T , uk(l()]d
is optimal
(Sk)x(X,t)j
(4.13)
Since
i=l,---, n, is given in
x,
From (4.7), (4.9), (4.12)
(Sk)x(x,t)l < C1E x 0
or since
< C1Sk(x,t) + C2(t+l)
is bounded uniformly on compact sets, (4.12) gives the
Sk(x,t)
required bound for
I(Sk )x
uniformly on compact sets.
For the "trivial" control
O, we have by (4.8) and. S O e
J(x,t,O) < B(1 + Ex
for suitable
B1 .
For suitable
M
When
u(T) - O,
li||t)
,
r = max(2m,;
o = I, we have
this implies
S
· r
C
Let us show that
Since
Since
satisfies the same inequality.
S(x,t)
)
i(T) = x + w(T).
we have
Sk(x,t) < J(x,t,O) <M(l + Ixir),
Hence
(O) = x, and
),T).
= UkP(k(g
This can be proved exactly as in [4, Lemma 3].
[5, Lemma 5.3].
x+ k
Lx[ik(),t-- ,uk (T)]drT
k = 1,2,---.
S
is
bounded below,
S(x,t)-+
Sk(x,t) = J(x,t; ukP),
as -lx
(4.8) implies
+
-
,
uniformly-for 0 <-t <--t 1 .
-13-
[ [
[Uk(T)
lEx
Sk(x,t) >
B2 t + ES.[ k(t)]
]d-
+[ik(T)[
0
X > 0
Given
R2 > R1
Let
by (4.6).
R1
there exists
- x It <R 2 - R1 }
{IIvkllt >
=
ll WI lt >
A3 =
II
with
lit
Al cA
U A
2
T)
For
.
P(A1 ) + P(A2 ) >
3
R 2 -R
.
IxI >R
3
S(x,t')
-
,
k
m
EB1
4t (R2
Ix]
as
Ix
=
i
uk(O)dO
P(A3)
<
and
hence
1
IUk(0) I2 dO
S0 [5k(t)] > A
and hence
-I
For
Ixl>R 2
~2
-
R 1) P(A2 ) + XP(A 1) - ( t
S (x)
on
Rn .
+
3)
Since the right side does not
This implies that
, uniformly for 0 < t < t
+
To obtain uniqueness,
p(x,t) + 0
large enough,
1
satisfies the same inequality.
S
as
vk(T)
Since
Kk(t) I >R
a lower bound for
depend on
,
(R2 - R 1 )}
R 1 ) P(A2 ) < tEx
A 1,
Sk(x,t) >
with
R 1 )}
From Cauchy-Schwarz
On
2
-
- x = vkT) + w(T), 0 < T < t,
4 (R2
Let
(R2
the sup norm on [O,t].
k(
S (x) > X
implies
and consider the events
A1 = {ik
A2
IxI > R 1
such that
p = exp(-S) is a
uniformly for
2,1
C21
0 < t <t
solution of (2.1), with
.
Since
V(x,t) is bounded
-14-
above, the maximum principle for linear parabolic equations implies that
p(x,t) is unique among solutions to (2.1) with these properties, and
with initial data
;O
0
p(x,O) = p (x) = exp [-S (x)].
Hence,
S
is also
unique, proving theorem 4.1.
c = constant
It would be interesting to remove the restriction that
made in this section.
5.
A lower estimate for
S(x,t)
.
To complement the upper estimates
in Theorem 3.1, let us give conditions under which
x|
-*Xc
(xf
at least as fast as
, m > 1.
section we make the following assumptions.
(5.1)
U
'
, ax.bounded, a
1
for some
r > O.
For each
(5.2)
and
g
go gx.'
i
Moreover,
gx
E
as
p(x,t).-In this
a E C2
with
*,n,
j=l,
j
Moreover,
Q.
1
' gx.x
gxr
For each t
j
Q.
For each
< m
are continuous on
4+
t
V
t
, V( ,t) E C
i j
V satisfies (4.4),
(5.3)
V
i
and
i,
r
t , g(.,t) E C 2 .
,
for
We take
X.X.
-+
This is done by establishing
0
a corresponding exponential rate of decay to
S(x,t)
V, Vx,
X.
1
Vx.x.
X.
.
1
E f
V,
v
r
X
are continuous on
Q .
We assume that
j
and that there exist positive
B--; M-such that
P0 E C2
(5.4)
exp [xixlm+l][p
Let
Theorem 5.1.
p(x,t) - 0
6 >0
Jx| - o
as
such that
Proof.
+ Ip
(x)+ Jp.O (X)l
2,1
be a
p(x,t)
() I]
solution to (2.1) such that
C
0 < t < tl .
, uniformly for
is bounded on
exp[61fxm+l]p(x,t)
M
Then there exists
Q .
Let
m+l
Then
(x,t) = exp [6P(x)]p(x,t).
,
P(x) = (1+1x2) 2
is a solution to.
f
'at =
(5.5)
axx+
2 tr
g = g-
g .
+ V'
x
6a x
V= V - 6g
·
+
(
x
2
x
(
a
Following an argument in [10], equation (5.5) with initial data
0
= exp(6P)p0
1(x,t) = Ex{
(5.6)
where
0(s.6)
[X(t)]exp
= x.
This solution
below.
data
It
[- lgdw
dX = Q[X(T)]dw,
X(0O)
Then
p
the probabilistic solution
-
1
Co-1-1 2 d
,VdT]
+
satisfies
X(t)
(5.7)
wit
6> 0
has for small enough
i
is bounded and
p = exp(-6)ir
, and with
is a
g
a
In the integrands
T > 0,
and
V
are evaluated at (X(T),T).
We sketch the proof of these facts
C2'1
2,1
C
1
p(x,t) tending to
solution to (2.1), with initial
0
as
Ixl + o
uniformly for
-16-
O< t < t
.
p = p
By the maximum principle,
which implies that
exp [6ixim+l]p is bounded on' Q
It remains to indicate why
properties.
We have
*x
i
is a solution to (5.5) with the required
E
e
1
(4.4),
a
is bounded,
there exist positive
and
-1
By assumption
.
V
satisfies
1j
'
gE
E9
al, a2
,
p < m
.
Hence, for
6
small enough
such that
2m
V(X,t) < a
Moreover, for some
2
- all
2mXI
K
1
2
|(5 (x)g(x,t)l < K(1 + jxlj 2),
< m .
From these inequalities one can get a bound
E(exp
for any
j > 0.
one gets that
, - gdw
la
g-2dt
dl + Vd t)]
This gives a uniform integrability condition from which
2,1
C2 '
ir is a bounded
solution of (5.5) by the usual
technique of differentiating. (5.6) twice with respect to the components
x 1 ,---,xn
Since
Corollary.
(5.8)
6.
of the initial state
x = X(0O).
This proves Theorem 5.1.
S = -log p, we get by taking logarithms:
For some positive
6 ,
6
S(x,t) > 61 x
i+l -
Connection with the pathwise filter equation.
signal process in (1.1) satisfies for
~ E C
The generator --A- of-the---- -,-
-17-
=
tr a(Cx)xx + b(x)-.x
xx
x
The pathwise filter equation. (1.5) for
(6.1)
Pt
=
AY =
y
(A
)p
p = pY
is
+ VY p, where
AQ - y(t)a(x)hx(xxx
·
)
VY(x,t)
.i
h(x)
2
1
y(t)Ah(x) + 2 y(t)
-
2
II(x)'a(x)hlx().
Hence, in (1.5) we should take
n
(6.2)
-b + y(t)ah x + Y,
gY
Yj
=
Da..
ax.
i=1
j
xa
vY
(6.3)
= vY
div(b - y(t)ahx) +
axax
axi ax
,j=
To satisfy the various assumptions about
above, suitable conditions on
cr, b, and
the local Ho3lder conditions needed in
continuous on
[O,t].
h
g = gY,
Y
V = V
must be imposed.
§4 we assume that
made
To obtain
y(-) is HI'lder
This is no real restriction, since almost all
observation trajectories
y(.)
are H6older continuous.
To avoid unduly complicating the exposition let us consider only
the following special case.
made for the existence theorem in
b, b x
§4.
We assume that
3
b E C
with
bounded, and all second, third order partial derivatives of
of class
2
for some
r.
Let
a polynomial of degree
L
, with
(6.4)
a = identity, an assumption already
We take
r
liirm
IXIc
h
be a polynomial of degree
h(x)[
=
,
lim S (x)
IXIfIc
= +c.
b
m and
S
Then all of the hypotheses in § 's 2-4 hold.
polynomial growth of degree
Vy
m-l
as
is the sum of the degree' 2m
polynomial growth of degree
SY = -log pY .
Let
M1 ,M2
depend on
t > m+l .
-
2
h2(x)
and terms with
< 2m.
From Theorem 3.1 we get the upper bounds
SY(x,t) < M2 (l+Ixlm+), 0 < t
(ii)
we need
polynomial
has
, while in (6.3)
X
SY(x,t) < M(l+]x[P), O < t< t
(i)
where
Ix +
In (6.2), gY
y
.
For
p
, P = max(m+l,£)
< t < t
1,
m > 1
= exp(-S ) to satisfy (5.4)
The Corollary to Theorem 5.1 then gives the lower
bound
SY(x,t)-> 6xlm +1
(6.6)
From (6.5)(ii) and (6.6) we see that
Ix
m+ l
, at least for
0 < t<
t,
in case
and
t
0 < t< t
SY(x,t) increases to
bounded away from
+X
like
O , and for
£ = m+l
q = exp(y(t)h)p
Finally,
any
m > 1
- 61
is a solution to the Zakai equation.
~ E Cb (i.e., Pcontinuous and bounded on
At(+): =
For
Rn ) let
(x)q(x,t)dx
Rn
At(+) = E [x(t)]exp
(h[x()]dy - - lh[x(T)] dT)l , (y)
0
where
E
o
denotes expectation with respect to the probability measure
P
obtained by eliminating the drift term in (1.2) by a Girsanov transformation.
The measure
At
t
is the unnormalized conditional distribution of
x(t)
By a result of Sheu [10] At=A t and hence
q(',t)is the density of
At
.
In fact, both. At , At are weak solutions to the Zakai equation. Moreover,
ot
o
t
0
0-
EAt(1 ) = 1, EA t ( l)
< 1
The inequality is seen by approximating
corresponding density
Akt .
Then
qk(x,t)
h
hk
by bounded
with
of the unnormalized conditional distribution
(see [10])
EA.t(l)
for any continuous
P
1,
Akt()+ At ( ) as k
with compact support.
Hence,
+ o
,
EAt ( 1) <1 . The
uniqueness theorem in [10] for weak solutions to the Zakai equation implies
A
t
=A
t
-20REFERENCES
1.
Baras, J., G. L. Blankenship, and W. E. Ilopkins, Existence, Uniqueness,
and Asymptotic Behavior of Solutions to a Class of Zakai Equations
with Unbounded Coefficients, Electrical Engrg. Dept., Univ. of Maryland, preprij.
2.
Davis, M. It.A., Pathwise nonlinear filtering, in Stochastic Systems
(ed. M. Haziwinkel and J. C. Willems) NATO Advanced Study Institute
Series, Reidel, Dordrect, June-July 1980.
3.
Davis, M. H. A., and S. I. Marcus, An introduction to nonlinear filtering,
ibid.
4.
Fleming, W. H., Controlled diffusions under polynomial growth conditions,
in Control Theory and the Calculus of Variations (ed. A. V. Balakrishnan), pp. 209-234, Academic Press, 1969.
5.
Fleming, W. H., Stochastic control for small noise intensities,
SIAM J. on Control 9 (1971), pp. 473-517.
6.
Fleming, IV.H., Stochastic calculus of variations and mechanics,
submitted to J. Optimiz. Th. Applic.
7.
Fleming, W. 11. and R. W. Rishel, Deterministic and Stochastic Optimal
Control, Springer-Verlag, 1975.
8.
Hopf, E., The partial differential equation
ut + uux = 1Ux'x
Com-
munication Pure Appl Math 3 (1950) pp. 201-230.
9.
Mitter, S. K., On the analogy between mathematical problems of nonlinear filtering and quantum mechanics, Richerche diAutomatica.
10.
Sheu, S-J,
11.
Sussmann, H.,