January 1982 LIDS-P-1174 OPTIMAL CONTROL ... - Semantic Scholar
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
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
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 R1
Let
by (4.6).
R1
there exists
- x It
=
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)
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 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
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)
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
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
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 R1
Let
by (4.6).
R1
there exists
- x It
=
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)
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 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
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)
