....~
STOCHASTIC. AND f:JULTIPlE HIE:'lER HlTEGRAlS
FOR GAUSSVm PROCESSES*
by
Steel T. Huang
Department of Mathematias
University of Cinainnati
Cinainnati~ Ohio 45221
and
Stamatis Carrbanis
Department of Statistias
University of North CaroZina at Chapel Hill
Chapel HiZZ~ North Carolina 27514
Institute of Statistics f!lim.eo Series No. 1087
August~
II
* T11is
1976
research was supported by the Air Force Office of Scientific Research
under Grant AFOSR-75-2796.
STOCHASTIC AND !\1JLTIPLE W1ENER INTEGRALS FOR GAUSSIAN PROCESSES*
by
Steel T. I-luang
Departnent of Mathematics
University of Cincinnati
and
Stama;is Carnbanis
Department of Statistics
Ur,iversity of North Carolina at Chapel Hill
I/.!Ultiple Wiener integrals and stochast::.c integrals are defined for
Gaussian processes, extending the related notic,lfis for the Wiener process.
It
is shown that every Lz-functional of a Gaussian process admits an adapted
stochastic integral representation and an orth')g.mal series expansion in tenns
of rnu1 tiple Wiener integrals.
Also some ::,esults ·Jf Wiener's theory of non-
linear noise are generalized to noises other than white.
* This research was
supported by the Air Force Office of Scientific Research
under Grant AFOSR-7S-2796.
I
o.
H1TRODUCTI Of!
Let us first fix our basic notation.. and tenninology.
We will consider
throughout a zero mean (for simplicity) Gaussi<m process X =
defined on a probability space
(Xt~
t€T)
T vlill be an interval of the real
(S1~B9P).
B is
line 9 even though more general index sets could clearly be used.
usually taken to be B(X)9 the a-field generated by the process X9 or B(X) 9
the completion of B(X).
a Gaussian process.
There are
tlr.rQ
important Hilbert spaces associated to
= LZ(S1 9B(X)9 P)9
The nonliaear space of X9 LZ(X)
consists
of all B(X)-measurable random variables with finite second moment which are
called (nonlinear) LZ-functionals of X.
the closed subspace of LZ(X)
The linear space of X9 H(X), is
sparnled by
~
9
t €
T~
and its elements are
called linear 1 Z-fUnctionals of X.
The first useful notion in the study of the nonlinear space of a Wiener
process is the Multiple Wiener Integral.
This notion \vas first introduced by
Wiener (1938) 9 who ten-.led it nPolynomial Chaos!; ~ and was redefined in a
somew~~t
deeper way by Ita (1951).
Ito showed that his TIID1tipie integrals of
different degree have the important property of
beL~g
mutually orthogol1a1 and
also presented their connection with the celebrated Fourier-Hermite expansion
A
of 1 Z-functionals of Cameron and Hartin (1947). Subsequently, Ito (1956)
extended to general processes with stationary independent increments the WienerIto expansion of 1 Z-ftmctionals in terms of multiple Wiener integrals. In his
inportant ""'ark on nonlinear problems Wiener (1958) reinterpreted the multiple
Wiener integrals for a Wiener process in an extremely simple and intuitive way
and made some interesting applications.
Finally Neveu (l96G) and Kallianpur
2
(1970) studied the cormection bet\'leen the nonlinear space of a Gaussian
process and the tensor products of its linear space, which sheds new light and
gives more insight on the structure of the nonlinear space.
The first objective of this "lork is to define multiple Wiener integrals
for general GaussiaY1 processes and to use them in extending iNiener's theory of
nonlinear noise.
The groundwork is in Section 1 l"1here the Hilbert spaces of
appropriate integrands for the nmltiple Wiener integrals are introduced and
studied.
The multiple Wiener integrals are then defined in Section 2.
Section'
S includes the extension of some basic results of Wiener's nonlinear noise
theory from noises generated by the Wiener process to noises generated
similarly by processes with stationary Gaussian increments (Theorems 5.1 and
S.2)? as well as a simple but interesting result on processes with stationary
increments which we could not find :in the literature (Lemma 5.5).
The second useful notion in the study of the nonlinear space of a Wiener
process is the stochastic integral.
The stochastic integral was first
introduced by Ito (1944) for the Wiener process and later generalized by Meyer
(1962) for martingales.
Every l Z-functional of a Wiener process has a
representation as a stochastic integral~ where the integrand is adapted to the
Wiener process.
T11e second objective of this work is to define a stocl1astic integral for
general Gaussian processes? and this is done in Section 3.
The general pro-
perties of the stocllastic integral are stated in Theorem 3.2 and some specific
stochastic integrals are calculated (Theorems 3. 5 and 3. 7) .
The differential
rule of the stochastic integral will be developed elsewhere.
The stocl1astic
integral is defined :tor' general ifitegrands, not necessarily adapt"ed or"'nonanticipatory.
In Section 4 it is shown that each Lz-functional of a general Gaus-
sian process has a representation as a stochastic integral where the integrand.
3
is adapted to the Gaussian process (Theorem 4.2)
0
The stochastic integral
of nonanticipatory integrand.s is also considered (Theorem 4.3), but
the LZ-functionals which have nonanticipatory stochastic integral representations have not been characterized yet.
It should be noted that the two representations of LZ-functionals of a
(general) Gaussian process presented here (the first as a series of multiple
Wiener L.'1tegrals and the second as a stochastic integral) open the way to the
study of nonlinear devices with (general) Gaussian inputs.
Finally a brief summary of some facts on tensor products of Hilbert spaces
ai'ld on Hennite polynomials is included in an appendix for ease of reference.
4
1.
THE IiI LBERT SPACES
11.2 (R)
Al~D
A2 (R)
Throughout this section T will be an interval? closed or open? bounded
or unbounded? and X = (Xt , te:T) a second order process \'dth zero mean and
covariance function R(t?s). Integrals over T will be denoted by the
integral sign with no subscript, and IE will denote the characteristic
ftmction of the set E.
It is shm'JIl in Loeve (1955, p. 472) that the following two integrals
Itf1
To
R
I ~LtJ~t
an~
=R
J(f)
I
f(t)xtdt
can be defined as the mean square limits of the corresponding sequences of
approximating Riemann sums if and
OI1~Y
if the following double Riemann
integrals exist,
R
aTlt
If
then I(f)
2
f(t)f(s)d R(t?s)
and J(f)
and R
II
f(t)f(s)R(t?s)dtds ?
are random variables with means zero and variances
the corresponding double Riemann integrals.
1.1. The Hilbert Spaces 1I.Z(R) and AZJBl
Consider the set SI
f(t)
= l~
f n lea b J(t)? (an?bn ]
n' n
I
SI
of all step functions on T?
f(t)~
c
N
= l:
1
T, and define
fn(~
n
is clearly a linear space and for all
- Xa ) .
n
f,g e: SI we have
5
.....
E
(f
E
f(t)dXt 0
f f(t)dX
I
=0
t
=
g(t)dXt )
II f(t)g(s)dZR(t~s)
where the double integral is defined in the obvious way.
Two step functions
f»g will be considered identical if
II
(f(t) - get)) (f(s) f~g €
If we define for
then
(Sr ~ <0~ 0»
51
•
~
is an inner product space.
bilinear and symmetric properties~
only when f
g(s))d2R(t~S) = 0
<f~f>
Indeed <f ~ g> has the ordinary
~ O~ and <f,f> = 0
= E (f fdX)2
is the zero element of 5 r according to the convention intro-
duced above.
Now let AZ(R)
be the completion of SI ' so that it is a Hilbert space
with inner product denoted again by <0 ~ 0>.
Cauchy sequence of step ftmctions.
treat elements in AZ(R)
11 f(t)g(s)dZR(t,s)
as
11
A typical element in
Az (R)
is a
However, "Ie will find it convenient to
formal' I functions in t
E:
T and to write
for the inner product <f~g> (see Theorem 1 for a partial
justification).
Notice that for
f
€
through its increments.
there is a point
to
e: T
the integral 1 f(t)dXt depends on X only
Thus we may suppose without loss of generality that
SI
such that
6
=0
X
.
t
a.e •
0
Under this assumption we can establish an isomorphism between H(X)
A (R)
Z
as follows.
and
The map
..
SI -+ H(X): f 1-+
f fdi
preBerves inner products and hence it can be extended to an isomorphism on
AZ(R) to a closed subspace of H(X). But the set
where
H(X)
It
= l(to,t]
and It
€
SI.
t ~ to and
= -l(t,t
] for t < to ' generates
o
It follows that the isoroorphism is onto H(X), Le.
for
A2(R)~
We denote this isomorphism by I
H(X).
and we define the integral of f
with respect to X (which we write as
view elements of AZ(R)
!
f(t)~
AZ(R)
following our convention to
€
as formal flmctions) by
f
f(t)dXt
= l(f).
The properties of this integral follow from those of I
and are the analogues
of the properties of the integral \'lhen X has orthogonal increments (see e.g.
Doob (1953)).
The integral is defined for I1ftmctions" in AZ(R)
and tlUls it
is of interest to identify usual ftmctions in A (R) besides the step funcZ
tions. Two such classes of functions are identified in the following.
7
Under the additional asstm:q)tion that R(t,s)
is of bomded variation on
every botmded subset of TxT, Cramer (1951) defined AZ(R) as the completion
(with respect to the same imler product) of the set S~ of all functions f
whose double Riemann integral R ii f(t)f(s)dZR(t,s)
exists.
However, Cramer's
definition is not appropriate for the general case (where R is not necessarily
of bounded variation on b01.mded sets) since then It may not be in AZ(R) and
thus AZ(R) may not be iSOlOOrphic to H(X). In this sense our definition of
AZ(R) is the appropriate generalization of the definition given by Cramer.
That
S~
is always (even when R may not be of bounded variation) a sub-
space of AZ(R) and that for
the fact that
£~S~
l(f) = R J f(t)dXt ' follow inunediately from
is equivalent to the existence of R J f(t)dXt and the
f~S~?
approximating Riemann sums for R J f(t)dXt are of the form J fndXt with
f n E Sl' It can be also shown that when R is of botmded variation on botmded
sets then the two definitions of AZ(R)
coincide.
We show instead the following
result which is more useful for our purposes.
R(t,s)
and points a
is said to be of bounded variation on [a,b]x[c,d] if for all N,M
= to<tl< ••• <t.N = b,
c
= sO<sl<"'<~~ = d
the sum
N. IV.i.
( tri ' s m ) .
(t' ,s') _
"
,
Lri~l m~l I~(t
s ) RI 1S botmded, where ~(t s) R - R(t ,s )-R(t ,s)-
n-l' m-l
'
R(t,s' )+R(t,s). Also R is said to be of bounded variation on every finite
domain of TxT if it is of bounded variation on every [a,b]x[c,d]
c:
TxT.
Such
an R deternrlnes uniquely a a-finite signed measure on the Borel subsets of TxT,
denoted again by R, such that R(Ct,t']x(s,s'J)
= ~~~:~)I)
R.
Let L1 be the
set of all measurable functions f on T such that the following Lebesgue
integrals are finite -
8
II
.If
for all
(a~b]
f' e A2 (R)
If(t) I l(a,b](S) d2 IRI (t,s) <
c T 9 l'v'here
that the function f
Z
If(t)f(s)!d IRI(t 9 s) < 00
IRI
00
is the total variation measure of R.
in
LI represents an element in AZ(R)
such tllat for all geSI ~
<f'9g>
Notice that if such an ft
We will then denote
fV
We say
if there is a
= If f(t)g(s)d2R(t~S).
exists it is unique since SI
by f
and we will write
f
€
is dense in
A2 (R).
Az (R) •
With this conven-
tion l'le have the follovnng
e
1}ll3ORE~
TxT.
Then
Let R(t,s)
1.1.
is a dense subset of Az(R).
L1
II Ifl(t)fZ(s)ldZIR(t,S)I <
00
<fl '£2>
PROOF.
prove that
r
AZso if fpfZ
IE
€
= If
AZ(R) 9 i. e. there is an f e AZ(R)
IncI~
n
L
1
and
2
f l (t)fz(s)d R(t,s).
= If
= 1,2 9".,
Then IE e LI and we will
such that for all g€Sr ~
2
lE(t)g(S)d R(t,s).
be a finite interval containing E (so that
always"fLTld
€
then
Let E be a bom.ded Borel subset of T.
<f,g>
Let
be of bounded variation on evepy finite domain of
IRI (IxI)
<
00 ) .
We can
with each In a finite union of half open
intervals such that
IRI ((In~E)xI)
+
0 as n~.
9
e
Since
it follows that <II
Define
A2 (R).
f
n
~lI
m
e I~
11
= 1, Z~ • • •
~
f
00
is a Cauchy sequence in
and thus {II }n=l
R(ExE)
by f
A2 (R)
€
We first sho\v that
I~
>
n
= lim
II
.
n
does not depend on the approximating sequence.
be another such approximating sequence and fY
= lim
Let
II' .
n
Then for each interval JeI we have
= lim 1<11
1<£-£', lJ>1
n
= lim
-II"
n
1J >1
IR(InxJ) - R(I'xJ)I
n
~ lim {IRI ((Inlili)xJ) +
IRI ((I~lili)XJ)}
~ lim {IRI(CInL\E)xI) + IRr(CI~L\E)xJ)}
If M
r is the closed subspace of AZ(R)
then it is clear from the above that
that
f
f,f'
l(a~b]'
generated by all
€
Mr and f-f'
= O.
.L
MI'
(a,b]eI,
It follows
= f'.
We now show that
E.
Let J
n
= 1,2,...
I~,J~ e InJ
and I n ,
.
n
Define
and
(I~L\E)x(InJ) c
f
does not depend on the finite interval
= 1,2,...
I~ = InnJ
J~
IRI((I~L\E)x(InJ)) ~ 0,
(InL\E) xI ,
(J~L\E)x(InJ)
= JnnI ~
n
= lim
n
= 1,2,...
.
containing
and In'
Then clearly
IRI ((J~L\E),(InJ)) + 0 since
e (JnAE)xJ and
the result of the previous paragraph applied to
lim II
I
have the same properties as
and
I
II' = lim 1J'
n
n
IRI
I,J and
= lim
1J
n
is a measure.
InJ we have
From
10
and thus f
does not depend
011
I.
Hence f is well-defined. Now fix J
interval containing EuJ and In
l<f~lJ>
n
~
<f,g>
=
= 1~2~ .•• ~
T and let I be a finite
as before, i.e.
II
Then
+ f.
n
- R(ExJ)I = lim IR(InxJ) - R(ExJ)I
lim
S
Hence <f,lJ>
= (a~b] c
IRI (CInll.E)xI)
=
o.
R(ExJ) and since J is arbitrary it follows that for all
g~SI ~
= II lE(t)g(s)d2R(t~S). Thus we have shown that IE ~ AZ(R).
Now if E and F are bounded Borel subsets of T, denoting by IE and IF
also the corresponding elements in
AZ(R)
we have
<lE~lF> =
I is a finite interval containing EuF and as before II
we have (since
R
+
n
R(ExF).
Indeed if
IE and 1J
+
n
IF
is syrrraetric)
IR(InxJm) - R(ExF)I s IRI(Inx(Jmll.F)) + IRI(CInll.E)xF)
s
and thus
<lE~lF>
IRI ((Jmll.F) xI)
+
IRI ((Inll.E)xI)
+
0
= li~ R(InxJrn) = R(ExF).
It then follows that if </> is a simple ftmction in LI with bounded support
then
ct>
~ A (R), and
Z
Now let
f~LI.
if </>1 '</>2 are two such ftmctions then
Then there exist simple functions <l>n' n
bounded support such that
gence Theorem that
I</>n l +
IfI
on T.
= 1,Z~ •••
, with
It follow·s from the Bounded Conver-
11
n , n:: 1,2, ••• , is a Cauchy sequence in AZ(R) and we denote its limit
Hence
<P
by ft.
Then for all
g€SI
<f t ,g> :: lim <<pn,g>
II
=
= 1imff
2
<Pn (t)g(s)d R(t,S)
2
f(t)g(s)d R(t,s)
again by the Bounded Convergence Theorem since
f€L r
and g€Sr
imply
II If(t)g(s) IdZ/RI (t,s) < 00. Since the values of <f t ,g> for g€Sr determine
ff
is uniquely detennined by £,
uniquely, the last equality implies that f'
independently of the approximating sequence <pn '
thus
L1
AZ(R).
It follows that
f € AZ(R)
and
Lr contains $r it is dense in AZ(R).
For the last statement of the theorem, with the obvious notation, we have
c
Since
<f1 '£2> :: linl
=
II
<~l,n,<PZ,n> ::
lim
II
<P1,n(t)<Pz,n(S)dZR(t,S)
2
f l (t)f (S)d R(t,S)
z
where the additional assumption on f1'£2
makes the Bounded Convergence Theorem
applicable.
0
Consider now the set SJ
integral
of all functions
R 1/ f(t)£(s)R(t,s)dtds
Two functions
f
f
on T such that the Riemann
exists and is finite.
SJ is a linear space.
and g in SJ will be considered identical if
R
II
(f(t)-g(t)) (f(s)-g(s))R(t,s)dtds :: O.
For f,g € SJ we define
E(I f(t)~dt
I
0
f(t)~dt
:: R I f(t)Xtdt
I g(t)~dt)
= R
II
and then we have
f(t)g(s)R(t,s)dtds.
12
e
Define for
f,g
€
SJ '
<f,g>
Then
(SJ' <0, 0»
= R II
f(t)g(s)R(t,s)dtds.
becomes an inner product space.
completion of the inner product space SJ
typical element in A2 (R)
is defined to be the
and so it is a Hilbert space.
is a sequence of ftmctions convergent in norm.
ever fonnally we shall treat elements in AZ(R)
ff f(t)g(s)R(t,s)dtds
A2 (R)
Again a
How-
as ftmctions and write
as the inner product <f,g>.
In order to establish an isomorphism between H(X) and AZ(R) we shall
assume that X is mean square continuous which is equivalent to the continuity
of the covariance function R(t,s).
n
Consider the sequence of functions
1
(t) where l' is an interior point of T. It is easy to show that
(1'-n'1']
this sequence is a Cauchy sequence in A2(R), whose limit is denoted by ( \ ' and
0
1
that
Then, the map
SJ
-)0
I~ I f(t)~dt
H(X) : f
preserves inner products and its range includes X-r
(which is linearly dense in H(X)
by mean square continuity).
onto H(X).
extended to all isomorphism on AZ(R)
morphism is denoted by J
and for
for all interior
f
€
AZ (R)
f f(t)xtdt = J(f).
l'
of T
Hence it can be
Thus AZ(R) "" H(X) , the iso-
we define
13
A useful cOIUlection between the integrals
and 1.2 can be established as follows.
to
Let
is an arbitrary but fixed point in T.
tinuous. )
f
€
= liD ~du
= J(l(t
(lCto,t] € A2(R)
' t])
o
where
since R is con-
A2(R)
H(X)
= /\.2 (r)
If X is mean square aontinuous then A2 (R)
and for
= /\.z(r),
f
Henae
Zt
and the spaces /\.2
Then
THEOREM 1. 2.
aZZ
and J
I
f(t)xt dt
= J fCt)dZ t
.
= H(Z).
We first prove that the existence of R IT IT f(t)f(s)R(t,s)dtds
equivalent to tl1at of R IT IT f(t)f(s)d 2rCt,s). We may assume that T is a
PROOF.
closed interval by the very definition of a Riemann. integral.
is
We also assume that
/f(t) I < M for all t€T; othelWise neither Riemann integral will exist.
Consider
the typical Riemann sums
RJ =
R1
LL f(t.)f(s.)R(t.
,so)
1
J
1 J
IA1 1 IBJol
0
= L L f(ti)f(Sj)r(AixB j )
{Ai}' {Bj } are interval partitions of T, ti€Ap Sj€Bj , and IAil, IBj I
denote the lengths of these intervals. By the unifonn continuity of R(t,s) we
where
have that for every e:>0,
conclude that
2
2
IR1-RJI::; M ITI e: as maxCIAil, IBj /)
-+
0.
We thus
14
R
II f(t)f(s)R(t~s)dtds = II
R
2
f(t)f(s)d r(t,s)
and the existence of one integral implies that of the other.
Note that
AZ(R) •
st
Thus
is dense in A2(r)
(since r is continuous)
In
short~
and
SJ
is dense in
AZ (R) = AZ(r) .
For a step ftmction f we clearly have I f(t)Y't:dt
is true for all f e: A2(r) by the continuity of I
=J
f(t)dZ t ; hence this
and J.
0
AZ(R) may contain interesting classes of functions larger than SJ. Let
L be the set of all measurable ftmctions f on T such that the following
J
Lebesgue integrals are finite
. -II
If
If(t)f(s)R(t,s) Idtds < 00
If(t)1 l(a,b](S)
IR(t~s)ldtds <
00
for all
(a,b] cT. We will follow the same convention (as for AZ ) in treating
ftmctions f in LJ as elements of A2(R) if there is a f' e: AZ(R) such that
for all g in a dense subset of A2(R),
<f',g>
=
II
f(t)g(s)R(t,s)dtds.
With this convention the following is a corollary of Theorems 1.1 and 1.2.
COROLLARY 1. 3.
subset of A2(R).
Let R( t, s)
be continuous on TxT.
AZso if fl~fZ e: LJ
and
Then
LJ
is a dense
If !f1 (t)f Z(s)R(t,s)ldtds < oo~
then
15
We now remark on the relation between A2,A 2 spaces and L2 spaces. The
spaces AZ(R) and Azert) are generalizations of L2 spaces. In general they are
larger than L2 spaces. As an example, consider R(t,s) a continuous covariance
function on [a,b]x[a,b] and let ret,s) be defined as before. Then AZ(R)
Every function f in LZ([a,b],dt) belongs to A2(R)
Corollary 3 since
Az(r).
II
If(t)f(s)R(t,s)ldtds
~
max IR(t,s)I
0
~
max IR(t,s)/
0
e
AZ(r) by
If(t)ldt)2
Ib-al
I
2
f (t)dt <
00
A2 (R) = A2(r)
for all interior points t of T
and similarly II If(t) II(a,b] (s)IR(t,s)/dtds <
is not in L2([a,b],dt) since ~ = J(Ot)
implies R(t,s) = II 0t(u)os(v)R(u,v)dudv.
(I
=
=
00.
~bwever,
0t
E
Nevertheless, there is a special case where A2(R) reduces to an L2 space.
Let X be a zero mean process with orthogonal increments. Asstmle Xt = 0 a.e.
o
for same fixed toET. Then, R(t,s) = F(tov(tAs)) + F(toA(tVS)) where F(t) =
EX~ if t ~ to and = -1:X~ if
t
~ to. F is nondecreasing and thus R(t,s)
is of bounded variation on every finite domain of TXT, and the associated measure
concentrates on the diagonal t=s of TxT.
In this case AZ(R)
= L2 (T,
dF(t)).
In particular, if X is the Wiener process A2(R) = L2(T,dt). A slightly weaker
result is easily seen to be valid when R(t,s) = I~ I~ k(u,v)dudv + tAs, with
k
E
L2(TxT); in this case the two sets (rather than spaces) are equal, AZ(R)
L2(T,dt), and their norms are equivalent.
general result.
In fact we have the following more
=
16
e
YriEOREI"l 1.4.
If R and 8 are tzvo equivalent oovarianae funotions (i.e.
the associated zero mean Gaussian measures are equivalent) then the following sets
are
equal~
A2 (R)
PROOF.
= AZ(S)
and AZ(R)
= A2(S)~ and their norms are equivalent.
Denote by llR and llS the zero mean Gaussian measures with co-
variances RandS on 80~T), and by X = {X 1 tlET} the coordinate process on
t
T
R • .An element in AZ(R) is really an equivalence class of sequences <t>n € SI
r
such that
<PndX is a Cauchy sequence in LZ(dllR)
"lith a common
limit~
and such
an element is denoted by <<t>n>.
AZ(R) and that llR:: 118 ' i. e. ~ llR and llS are
We ShOl'l <CPn> € AZ(S) . First observe that <¢n> = <<t>n t > for all
Suppose that <<t>n>
equivalent.
subsequences
e
LZ(llR!
{n'}
and a.e.
€
r <t>ndX is Cauchy both in
It follows from llR:: llS that r <t>ndX is also Cauchy
is Cauchy in Lz(dllS)' since each r ¢ndX is a Gaussian
of {n}.
[llR]'
a.e.
EllS]
and hence it
r.v.
This implies that
follows from symmetry.
<<t>n>
Thus we may assume that
€
AZ(S).
The case of A2
So AZ(R)
C
A2(S)~
and the assertion now
spaces is shown similarly.
0
It should be remarked that the converse of Theorem 1.4 does not hold.
exarl1ple~
AZ(R) = AZ(a,R)
are not equivalent
For
as sets and their noms are equivalent but R and a,R
(a,;tl).
1.2. Tensor Products of
AZ(R)
and Az..ilil
We now study the tensor product spaces J'AZ(R) and J'A2 (R) • Consider the
set sip) of all step functions f(tl"" ~ t p ) on TP. Define the following
function on
sfp)xsip) ,
17
<f~g> = II··· If f(tl,···~tp)g(Sl,···~Sp)d2R(tl,Sl)···d2R(tp~Sp)
4It
<f-g~f-g> = O. Let II IX"'x I
' 1
P
Jlx ••• xJ
with g if
and identify f
,
€
S(p)
p
~
(Le.
Ip J
are bounded half open intervals in T).
j
<II x
xl ~lJ x
xJ >
1 ... p l ' "
P
= II
2
II II (t)lrP (s)d2R(t~S)
I I p
1
This ir:iplies that
by
L"l
Then
II (t)IJ (s)d R(t,s) .•.
= <I Il ,lJ
= <II
I
>Az(R) ... <lI ,lJ >1\2(R)
p
@••• 0l
1
(Sfp), <0,0»
p
' lJ @••• 0lJ > D
1p
I p 0'- 1\2 (R) •
o
is an inner product space and we shall denote
1\2(~)R) the completion of sf!'). Since {II @. H@l I } is a complete set
D I p
@'-
1\2 (R) $ 't:iTe have
1\2 (#R) ::'... ,J)1\2 (R) .
e
A2(~R)
can be defined in a similar f!13I'.ner.
Let sJp)
be the set of
functions of the fonn
£(tl, .•• ,tp) =
where the
cP's belong to
SJ'
¥ ~fk)(t)1
Ie=l
S}p)
..• <pp(k)(tp )
-
is a linear space.
Define on S(p)xS(p)
J
J
the function
and identify
f
",lith IT
if <f-g,f-g>
= O.
With the observation that for
18
e
<cj>l Ct l ) ... 4>p Ct p L lPl (t l ) ... 1/Jp (tp )>
=R
II cj>ICt)1/JICs)R(t~s)dtds
= <cj>1~1/Ji>A2(R)
.•. <cj>P~1/Jp>AZ(R)
= <cj>l~···~epp,
and with the fact that
that
(sJp), <o~o»
1/JI~···@1/Jp>
{epl~" .@cj>p}
is an :inner product space and that its completion, which is
As in the case of the spaces
and A2(~PR)
e
A2 (~R)
J>AZ(R).
AZ(R)
and A2(R)
we will treat elements of
as tlfonnal il functions and we will write the inner products
in a formal integral form.
and
~
A2(R)
is a complete set in ~A2(R) ~ we conclude
denoted by A2(~R), is isomorphic to
AZ(JJR)
n
~
As before, under some conditions, elements of
will be representable by functions on
'tl
AZ(ePR)
in the corresponding
sense and in this case we will identify the elements of AZ and A2 with the
functions (see Theorem 1.5 and Corollary 1.7).
The important point here is that
lve have identified tl1e abstract tensor product spaces ePAZ(R)
the (nearly) function spaces AZ(~R)
distinction between ~AZ(R)
and
A2 (~R).
and J'AZ(R)
From no,,; on we will make no
and A2(~R), and between ~A2(R)
and A2(~R).
Let R be of bounded variation on every finite domain of TxT and let
be the set of all measurable functions
f
on TP
with
Lip)
such that the following
Lebesgue integrals are finite
If ••. fJlf(tl, ••• ,tp)f(Sl~ •.• ,sp)ld2IRI(tl,sl) ••• d2IRI(tp'sp) <
00
2
ff ••• fflf(tl, .•• ,tp)llIlX ••• xlp(Sl"",Sp) d2 IRI (t1 ,Sl)· •• d !R! (tp'Sp)
for all bOlmded half open intervals
II' •.• ~ I p cT.
<
~
The following theorem can be
19
tit
proven lH:e Theorem 1.1 emd· thus .its· ·proof.. is ·omd:t1:ed.
THEOREM 1. 5.
Let R(t,s)
be of bounded va1'iation on every finite domain of
is a dense subset of A2(e~). A"l-so if f1'£2 E: Lip) and
2
2
II ... If If l (t1'''' , tp)fz(t1"'" t p ) Id IRI (t1' Sl) ... d IRI (tp'sp) < oo~ then
TxT.
Then
Lip)
<fl '£2>
= Jf···Jlfl(tl,···,tp)f2(Sl,···,Sp)d2RCtl,Sl)···d2R(tp'sp)
•
Theorem 1. 5 may be used to derive the well known fact that
L2(TP,dtP)
= ePL2(T,dt).
1HEOREM 1.6.
PROOF.
tit
If R(t,s)
is oontinuous on TxT" then
= A2CePr).
AZ(ePR)
This follows immediately from the facts that the set
{</lIe•.• e</lp' </liE:SJ }
is complete in both AZ(sPR)
and A2cJ'r), and that the two
inner products are identical on this set.
Let
LJP)
0
be the set of all measurable flmctions
f
on TP
such that the
following Lebesgue integrals are finite
JJ ••. JJlf(tl, •• ·,tp)f(Sl, ••• ,Sp)R(tl,Sl)· •• R(tp,Sp)ldtldsl···dtpdsp<oo
fJ... ffl£(tl,.·.,tp)llIlx .•• XI (sl,···,Sp)IRCtl,sl)···R(tp,sp)ldtldSl··dtpdsp<oo
p
for all bounded half open intervals
II' ... ,Ip cT.
With the usual corresponding
convention the following is a corollary of Theorems 1.5 and 1.6.
COROLLARY 1. 7.
subset of A2 CJ'R).
Let R(t,s)
be oontinuous on TxT.
A"l-so if £1'f2 E::iLJP)
R(tl,sl) ••. R(tp,sp)ldtldSl ••• dtpdsp<oo~ then
Then
LJP)
is a dense
and 11.• .11IflCtl' ••• ,tp)£2(sl""'Sp)
zo
Finally let us consider the symmetric tensor products
For f
€
....,
1\2 (~R)defme
f(t p ' .. , t p )
over all permutations 7f
= (7f l , .•. ,7fp)
syrrmetric version of f.
1! is first defined for f
II f liZ
p ~ pIli f liZ
1\Z(~ R)
Az (~R) •
f
= (pI) -1
'\ f(t ,. ~ ~ ,t7f) where t he sum 1S
.
7fl
7f
...,P
(l, ... ,p), and f is called the
of
L
is well-defined since
€
s~p)
?
f
is a 91functionli.
Indeed,
and then, using the easily verified fact that
, the definition is extended by continuity to
1\Z(sPR)
If f=1 then f
is said to be a syrmnetric ilfunctionVi •
be the subspace of all symmetric Hftmctions Vi in 1\2 (sPR)
show that 1\Z(ePR)
and ~AZ (R) •
# 1\Z (R)
0
Let 1\2 (iPR)
Then it is easy to
is a Hilbert space and iPAZ(R) ~ 1\2(iPR)
under the corres-
..., .. o@f
"" ~ (f (t ) o. of (t) )""
r.,:pR) be the
flo
Similarly, let A2\.@
l. l
p
p p
subspace of all symmetric "ftmctionsit in AZ(#R) • Then l'le can show that
pondence
0
;PAZ(R) ..., Az(iPR)
(under the natural correspondence).
identify @PAZ(R)
with 1\2 (iWR) , and &PAZ(R)
As before, we shall hereon
with AZ~R).
1.3. Fourier Transform on 1\zJ0PR) and AzC8PRl
Consider the covariance function R(t,s)
of a zero mean, mean square
continuous process X = OCt' -oo<t<oo} with (\ride sense) stationary increments.
For convenience such R is said to have stationary increments.
Let
(1.1)
Then it is well l<nown (Doob (1953), p. 552) that R(tl,sl; tz,sZ)
have the following spectral representation
and xt-Xs
21
(1.3)
is a finite measure on B(R) and V = {V~~ -OO<A<OO} is a process
with orthogonal increments and EldVAI2 = dF(A). We remark that
where dF(A)
(1.4)
H(~X)
where
~x
= H(~V)
denotes the set of increments of the process X.
Define the Fourier transform of f
sip) by
£
.:;. ..'. (~l-' .. ~~) -_If. :~. f ei(i:;l A1+·· .+tpAp)' f(t.
00
p " . ,tp)dt l ... dtp . ' .
The program is to define the Fourier transform f of every f in
(For this reason~ it is convenient to extend
€
<f,g>
sip),
un-
11.
2
(w~ £~J
2
from a real Hilbert space
2
From (1.2) it follows easily that
to a complex Hilbert space.)
for f
11. (sPR)
where the measure ~ is defined by d~(A)
= <f,i>
11. (tJiR) •
f
€
L2(RP~ ~p),
= (1+A 2)dF(A),
and
P P • Since the map F: S~p) ~ L2(RP~ ~p) : f I-+-
L (R
2
~].l )
f
is linear and preserves imler products, it can be extended to an isomorphism on
11.
2(~R) • We now shmv that F if onto 12(RP , ~p) • It is sufficient to show that
A
A
1(a1,bl]x••• x(ap,bp](A1'···'~)
= 1(a1 ,b1](Al) •.•
form a complete set in L2(RP,~P); or equivalently that
eibA_eiaA
l(a,b](A) =
iA
A
form a complete set in
L2(R,~).
Since Lz(R,dF) -
H(~V)
under the correspondence
22
h
+-+
r hdV,
it follows from
z~
(1.3) and (1.4) that {(eitA~eisA) .(li~) , s<t}
is complete in 1 2 (R,dF), and hence
"
{l(a,b]' a<b}
is complete in LZ(R,l.l).
We
thus have the following
'IHEOREM 1. 8.
The map F:
sip)
to an isomo:ephism from AZ(~R)
-+ L (RP ,l.lp):
2
."
f 1-+ f
has a unique extension
onto L2(RP,l.lP).
The extended map is again denoted by F and is called the Fourier transform
f
on AZ(9PR). We also write
for F(£).
If we let X be the Wiener process, i.e.,
LZ(RP ,dtP).,
dl.l(A) = ~'IT dA.
Therefore
R(t,s)
= tAs,
then A2 (sPR)
=
F reduces to the ordinary Fourier trans-
form on L2 (i~P ,dtP ) •
is stationary (which implies that R has
Suppose now that R(t,s)
ary increments).
station~
Then by Bochner t s theorem we have
R(t,s)
= Jei(t-s)A
with v a finite measure on B(R).
dV(A)
It is plain to deduce from (l.Z) and (1.5)
that
dV(A) = ~ dl.l(A) •
A
Thus AZ(#R)
AZ(~R).
f LZ(RP,
Let ret,s)
2
dP (A V(A))).
= f~ f~
NQltl
R(u,v)dudv.
we define the Fourier transform on
Then the covariance
r
has stationary
increments and AZ(J'r) = AZ(sPR) • The Fourier transform on AZ(qJJR) is defined
to be the Fourier transform F on AZ(~.lr). It is a simple matter to verify that
the spectral measure of r
is
(l+AZ)-ldv(A)
and thus
AZ(9PR)
f.
LZ(RP,vP).
23
e
THEOREI'vl 1.9.
If
RCt~s) is a aontinuous stationary aovarianae funation$
L1 (RP) is a dense subspaae of AZ(@PR), and F restriated to L1 CRP) is
then
the ordinary Fourier tronsfom.
PROOF. Let f
L (I~p) •
1
€
Then
fI···IIlfCt1~···~tp)f(Sl~···~Sp)R(tl~Sl)···R(tp~sp)ldtIdsl···dtpds p
~ v(R)PllfIl Z
p <
00
Ll(R)
since
IR(t~s)12 ~ R(t,t)R(s,s) = vCR)2. Si~milarly~ the second condition in the
definition of
LjP)
sfp)
is dense in
L"'1
C
L1 (RP)
is verified. Thus f
A2C~R) by Corollary
€
Azc@Pr) = A2C0PR)~
Since
1. 7.
it follows that Ll(RP)
is dense
A CJ>R) •
Z
To prove the second assertion it is sufficient to prove that for f
A
the ordinary Fourier transfonn f
A
IIfll
A (Q1)PR)
2
1.7 that
IfI
•
11£11
2
~
11£11
L (Rn )
A
belongs to L 2 (RP ~ vP)
implies
f
€
and
II fll
€
Ll(RP)
Z
Z pD
L (R
=
~ V'" )
L (RP , vP) • We have from Corollary
2
l
n
~2(0"R)
= ff .•. fri(tll •.. '~p)f(sl,···,sD)R(tl~Sl}••. R(tp'Sp)'
.
.I:
•
dt1ds1o .• dtpdsp • Substituting R and interchanging the order of integration
by FubinP s theorem, \'1e obtain
complete.
II £11
2
A (~R)
Z
= II fll
A
2
0
p
L (R~ ,v )
2
The proof is now
o
Let R be again a covariance function having stationary increments and let
f € AZ(0PR) • We define the translation f T of f by l' = (1'1' ••• ' Tp) as
follows. Pick a sequence of step ftmctions CPn such that lim CPn = f in
n
AZ(0 R). The translation of each CPn is defined by CPT~n(tl~ ••. ~tp) =
2
¢n(tl+Tl,···~tp+Tp). Clearly IICPT n l IZ n - I ICPn l 1 p • TIlis implies
'A2(~R)
A (0 R}
Z
24
that {4> T, n} is a Cauchy sequence in
lim
<P
T
,1n'
A_
(J>R),
._~
and f T is defined to be
A sinp1e argument shows that the definition of
on the choice of the approximating sequence
translation if f
{<I>n}
becomes the usual
is indeed a function.
l\ijlen R is stationary, the translation f T
= A2(@Pr)
similarly (or via the identity A2CePR)
THEOREM 1.10.
and f
f T does not depend
of f
EO
A (JJR)
2
can be defined
).
If R is a continuous covariance with stationary increments,
then 1\.2(~R) is invaraiant under transZ:ations" and for
f,g
EO
Az(#R)
we have
and
PROOF.
f
EO
Since R has stationary increments, all these assertions hold for
s~p). Hence they hold for all f
f:
1\.2 CJ'R)
by continuity (cf. Theorem 1. 8). L
TIle corresponding theorem for translations on A2 (JlR)
also holds.
25
2.
Suppose, X =
(~,
riULTIPlE UIEf"lER INTEGRALS
tE:T),
T an interval, is a zero mean Gaussian process
with covariance function R(t,s).
We shall define the multiple Wiener integrals
(I'M1 ' s) of the following two types:
J f(tl,···,tp)dY~l
J f(tl,···,tp)Xtl
where p = 1,2,... .
~
o
=0
while dealing witJ'1 integrals
(J):
I
P
t
P
••.
We will aSSlUl1e
(I):
The UN1
... dX
a.s.
for some to E: T,
I p ; and while dealing with integrals J
p
'
X is mean square continuous.
has been defined for X a Wiener process in It6 (1951) and
Wiener (1958); in this case f
is taken to be a ftmction in LZ(TP, dtP)
(P!f~Ip is an isomorphism on L2 (TP, dtP)
and
(the Hilbert space of all symmetric
ftmctions in L2 (TP, dtP) ) into L2 (X) • The rnaj or step in generalizing the
notion of the 1'1'11 I p to a Gaussian process other than the Wiener process is to
determine a proper Hilbert space of ftmctions on which I p will be defined.
Clearly II should be defined as the isomorphism I from A2 (R) onto
H(X). Now for p > 1, in accordance with the Wiener process case, it is reasonof the form </>1 (t1) .•• </>p(tp L
</>i E: !I.z (R), are admissible integrands, and their integral I p (f) is the iterated
able to expect that ftmctions
integral
1(</>1) ••. I (</>p)
f(tl"'" t p )
when </>1' •.• ,<I>p are orthogonal.
This suggests that
26
e
the Hilbert space
1\2(~R)
is the proper class of integrands for the MWI
Similarly the Hilbert space A2 (eI'R)
UtilI
Ip •
is the proper class of integrands for the
Once the classes of integrands are detennined ~ I,·ItVI' s can be defined in
Jp •
a straightforward manner.
We, will use the following result on the structure of the nonlinear space of
X (see Neveu (1968), Ka1lianpur (1970) ).
a Gaussian process
ze~o
Let X be a
mo~phiBm
<P
fl'om e
mean Gaussian proaes8.
""
Then there exi8ts a unique iso-
(where the spaae for p=O i.s the set of aU aon-
> -H0p (X)
p-U"
8tant r.v. 's in L2 (X)) onto LZ(X)
suah that
'"
cp(e8~) = e~-~II~II
;e-
where e '"
~
= l.p~O(pl)
-k ~
2~-.t",
'"
*PI""
'" 0pk)
cp ( ~l 8 ... 8~k
~
E:
2
If ~P". ~~k E: H(X)
H(X).
are orthogonaZ then
""
where p
= Pl+ ••• +PJc.
= (pi) -!.c~
"
PI' II ~111
If
{~Y~
YE:r}
2(~1) ... H
Pk ~ II ~k II
2(~k)
linearZy ordered) is a CONS in H(X)
(r
then the fami Zy
""
ep
""
ep
(P
,pi
I )~cp(~ YI;.•. ;~ Yk) =
....
py.J
Yl
Yk
y
1
k
t~y.. If'~ ,
(P : ,0 -~H
2(~y ) ,
Yk
'Py' ,II ~y II
k
,I
1
,k
k
p~O, ~1, Py1+ ... +PYk = p~ Yl< ... <Yk ' is a CONS in L2 (X).
Py ,II ~y II
.1.
vIe now define
Ip
~
p~l.
isomorphism I: f 1-+ ! fdX,
2(~y
)
I
Since 1\2 (R)
1\2 (~R)
""
Denote this isomorphism by 1*1>.
is isomorphic to H(X)"" under the
= &P1\2 (R)
is isomorphic to
ffP (X) •
For <PI' ••• ,<Pp orthogonal in A2 (R)
(
.
we have
27
whi:h suggests the following definition of I p : 1I.2(~R)
~ (IfP (X) ) ),
Furthermore we define
tensor of f
= Ipcf)
Ip(f)
for
(Le., the projection of f
f
€
-)0
(in fact onto
L2 (X)
f is the synmetric
II. (J'R), where
Z
onto the subspace of syrrmetric ten-
sors). The follOll1ing results are then immediate consequences of the fact that
....
I~ is an isow.orpmsm.
Let X be a zero mean Gaussian proaess satisfying
TI-IEOP.EM 2.1.
the 1',1WI' s
p~l) have the fo ZZowing properties
1p '
1p(af+bg)
= alp (f)
(f) g
+ b1p(g) , a,b
(I).
Then
11. (J>R) )
€
2
E
R,
1p(f) = Ip(f) ,
<Ip(f),Ip(g»LZ(X)
= p!<f,g>
<Ip(f),1qCg»LzCX)
=0
""p
I p (4>1e
where
{</>l, ••• ,<I>k}
I
""
10_••• -e4>k0J>k)~
8 of x"
8
= fee)
+
8
= H
PI' II 4>1 11
E
,
if p~q ,
I
z( 4>ldX) ... Ii
Z( </>kdX) ,
Pk' I I</>k I I
is an orthogonaZ set in AZ(R)
every LZ-funationaZ
and if 8 - £(8)
p
A (e R)
2
and Pl+.'.+Pk
= p.
LZ(XL has an orthogonaZ deveZopment
P~l IpCfp ) ' f p
= rp~lIp(fp) = lp~lIp(~)
E
then
II.ZCePR) ,
f p = ~, p~l.
AZso
23
In exactly the sane way we can define Jp(f)
restricted to
A2 (01)R)
:or'
f
€
n
A2(~R).
and
(p!) -~2J p
~ (H~ (X) ) • The corresponding
is an isomorphism onto
s J p ~ p~l.
It should be noted that the B.HI's I)(f) = f ... ff(tp •.• ~tp)~ ... dXt
and
I
1
P
Jp(f) = f .• .Jf(tl' ... ~tp)Xt ,,,Xt dtl ... dtp are defined in the mean square sense
~
1
P
and it is thus of interest to see whether they can be evaluated from the sample
Theorem 2.1 also holds for the
paths of X.
~1r:H I
Tnis can be done as follows when X is mean square continuous.
i;i = f<Pi (t)~
{i;i}f=l be a CONS in H(XL and write
{</>ifi=l
where
martingale convergence theorem we have for each f
A (~R)
€
2
We also have
00
r
<f,
il, ..• ,ip=l
·=r.
.
11,···~1p
and thus
and
Hence we have
€
A2 (R).
Then
Since clearly B(X) = B(~, n=1~2, •. ), by the
is a CONS in A2(R).
£ =
<Pi
~. 0..• 0~. ></>. 0 ..• 0</>.
~
11
a.
~
11
. <P.· 0 .... -0</>.
1l'···'~
11
lP
that
Let
29
n
1
!. _
Ip(f) = lim (plr~ .
(2.1)
11"'0,~-1
n~
{~i}
Now by choosing a CONS
paths of X, (201) expresses
the sample paths of X.
a.
.~. e ... ;~.
a.s.
11,oo.,1p 1 1
lp
such that each
~i
can be obtained from the sample
Ip(f) as the aos. limit of rov.'s obtainable from
One such choice is the following
0
Consider a sequence of
refining partitions of the (closed) interval T whose mesh goes to zero (eo g. ,
the dyadic partition) and denote by {til
the correspondir.g sequence of points
Then Xt . = J 1 (to' t . ]dX and by orthononna1
1
lizing {\.} "Te obtain the CONS {~i} where ~i = l<P i (t)cD<t and each <Pi is
of T, which form a dense subset
0
1
a step ftmction (a linear combination of let t ] ' 000,1 (t t ] ) ' Clearly each
0' 1
0' i
~i ca.'1. be obtained from the sample paths of X (as a linear combination of
\1
e
o'Xti ).
For the l'l'JI J p we have similarly an expression like (2.1) ("lith AZ(R)
playing the role of AZ(R) ) and the most natural choice of CONS {~i } in this
case is the following. Let {gi} be any complete set in LZ(T,dt) and defble
po
0
ni = Jgi (t)~dt as a sample path integral of a measurable modification of Xo
TI1en the sequence {ni} is complete in H(X) and by orthononnalizing it we
obtain the CONS
<Pi
{~i}
where
~i =
J<Pi (t)Xtdt as a sample path integral and each
is a linear combination of gl'"
,gi
If the eigenftmctions of R(t,s)
known one may then take <Pi to be the i th eigenfunction.
0
0
are
30
3•. STOCHASTIC WTEGRALS
In this section we let again X = (X , te:T), T an interval, be a Gaussian
t
process with mean zero and covariance function R and we shall define integrals
of the fOTn1! f(t)dXt with ret) a stochastic process appropriately defined.
This kind of integral l\laS first defined by It6 (1944) for X the Wiener process.
He observed that if the process
f
is adapted to X and if the approximating
r f (t
- X L to<t <. • . <tn '
l
k+l t k
then the usual argument in definin.g integrals can be carried through. That
Riemann sums are taken to be of the forms
1 )
1.(
(Xt
Wiener process is a Gaussian martingale suggests possible extensions of It6's
integral to martingales
~~d
to Gaussiml processes.
The stochastic integral for
martingales was successfully defined by Meyer (196Z) and thoroughly studied by
Ktmita and Watanabe (1967).
tit
The idea involved remains the same.
But in order to
extend Ita's integral to general Gaussian processes one should take a rather different approach using the tensor product structure of nonlinear Gaussian spaces.
He first generalize the notion of AZ spaces and define an integral denoted
by f f(t)0~. The details are omitted since the argument is analogous to that
in Section 1.1.
Let HI and HZ be Hilbert spaces.
an interval T, and let
R(t~s)
Let
~
be an HZ-valued function on
= <~ ,XS>H
• 'TIlen R is a nonnegative definite
Z
function (i. e., a covaria.'YJ.ce ftmctiOll). Consider the set Sr ;H of all Hl l
valued step functions on T, f(t) = I~ f n l(an,b ] ' (an,bn ] c T, f n e: HI •
n
SI;I\ equipped with the binary ftmction
<f,g>
=
II
<f(t),g(S»P1 dZR(t,s)
31
is an inner product space.
The Hilbert space
A ' H (R)
Z~ 11
is defined to be the
completion of SIoH. . Note that AZoH (R) reduces to
9 1
9 1
For f = I~ f n l(a b] € SloB' define
n' n
' 1
I
(3.1)
N
f(t)0 0X =
-"t
I1
f
n
AZ(R)
when
HI = R.
bn -Xan).
0 (A
T'nen
(3.Z)
is a norm-preservi11g linear map since
III
NN
Z
f0dXII H n,U = I I <fn,fm>u
1 1
I .-II
1'Qi£A2
,::;/II
/
Thus
I
<xb -xa ,
n
n
Xb
2
<f(t) ,f(s»q d R(t,s)
"1,
-x
>13
m am nZ
= II fll;
~
_ (R) •
JlZi1\ •
~
has a unique extension to an isomorphism . on A .H. (R) into H18HZ .
Z~ 1
0
It is clear that the range of 1 is r'110H(I1X), where H(I1X) denotes the closed
subspace of HZ
spanned by the increments of X.
We remark that Theorem 1.1 is valid for the present general case with the
proviso that one should read absolute values and usual products as norms and
inner products respectively.
increments and dF(t)
Also, if the HZ-valued ftmction
= IldXtll~2 '
sequel.
then
AZ;I\ (R)
= LZ;H
(dFL the Hilbert space
1
of all dF-square integrable HI-valued fmctions (for integration of H-valued
flmctions see e. g. Lang (1969) ).
then
X has orthogonal
The following simple fact will be used in the
32
(3.3)
f~®H2
(3.4)
AZoH (R)
?1
PROOF.
=
=
$
p~O
$
p~O
(G ®HZ)
P
~
AZor. (R) .
?~p
(3.3) is clear, and (3.4) follows from the following facts
o
Now let X = {Xt , tET} be a zero mean Gaussian process "'lith covariance R~
a.."'1d assume as 1.."1 Section 2 that Xt = 0 a. s. for some to € T. Let HI = l Z(X)
o
and HZ = H(X) (= H(M)). Then for f € AZ;1 (X) (R) the integral I@(f) =
2
! f(t)~~ is defined and belongs to l Z(X)@H(X). The program is to identify
as ma.~y elements in LZ(X)®H(X) as possible with elements in 1 (X) through a
Z
suitable (mbounded) linear map If and then define the stochastic integral of f
with respect to
(3.5)
X by
l(f) =
I
f(t)dXt = If(f
f(t)~)
•
Note that f?A ;L (X) (R) nay be vie"wed as a V1 second order stochastic process", and
Z
Z
each f(t) as an ilLZ-ftmctionaIV I of the entire process X; thus such Ps need
not be nonanticipati"'lg functionals of X.
We first define for each
p~l
a bounded linear map
(3.6)
as
fOllo~s.
Sp =
Pick.....a CONS
{~y'
{[~~l•••••I;~kle~ : I"'~,
y€rL
r
linearly ordered, in H(X).
Then
Pl+ ••• +Pk=P; Y'Yl""'Yk<r; Yl<" '<Yk} is a
cOlrrplete orthogonal set in H@P(X)sH(X).
Define '¥
p
on S
p
by
33
(3.7)
II?; 0t;
PI!.· .Pk 1
2
1/ '"
p y I-fP(X)@f-!CX)
= --p"T"J--';"-
PI!'· .Pk I
2
II 'lip (c:_@';y)/I;P+I=
-P
It
H
p!
if
y~yo
J
CX)
follows that
for some q
{2~
.•• ~p+l}.
Note that all elements on the right hand side of (3.7)
'"
fonn a complete orthogonal set in H@p+l (X), and for each such element there are
k
€
or k+1 (=:: p+l)
for some j
Y=YJo
elements in Sp corresponding to it depending on whether
or not.
It is now clear that '¥
,."
p
can be extended uniquely to
....
a bounded linear map with nonn p+l from tfI'CX)0HCX) onto H*p+ICXL and that
its definition is independent of the choice of a CONS in H(X).
that given any /;;p+l
€
H@P+I(X) one can find
rp
€
It is also clear
H@P(X)@H(X) such that
= /;;p+l and
'¥pCrp)
(3.8)
Notice
that~
since 'lip is a many-to-one map, (3.8) need not be true for all
,."
np
€
I{8P(X)@H(X).
34
Now let
Il'O
be the natural isomorphism between RI1.OH(X)
and H(X)
'"
(al1.O~
-+
into
a~).", We then define Il'* = t&P2: 0Il'p to?: the map from {$P2:otfPeX)}0H(X)
0
whose restriction to each H P (X)@H(X)
$ .... >lH@P(X)
p-
I Ill'pl I = p+l
is unbmmded in P~
Il'*
is
Il'
p
•
Si.llce
is an unbounded densely defined linear map
with domain
H* =
{¢€{p2:0
• H~(X)}~H(X): p2:0
t I Ill'P (¢P)1 12<oo}
,.,
where
¢ =
tP2:0¢p
easily seen that
~
p being the projection of ¢ ",onto HI1.Op(X)~H(X).
<P
H*
is a dense subspace of {$ >Or-fPeX)}@H(X).
p-
It is
Also it follows
,.,
is $p2:1~P(X).
from (3.8) (left hand side inequality) that the range of W*
Since $p2:0H;PeX)! LZ(X) ~ we have
'"
{6l
p2:0
denoting this isoTnorphism by
we define
Il' on H to
cD
h,0peX) }@H(X)~O 1 (X)@H(X)
2
~O.
l (X)
Z
Finally we let
H = @O(H*)
c
l Z(X)8H(X)
and
by
e3.9)
Then the stochastic integral is defined by
ll'(!f(t)@dJ(t)
= ll'oI@(f)
The set of all such
for all
f
€
f~s~ denoted by
(3.5)~ i.e.~
AZ;1 (X) (R)
z
!f(t)dY't = 1(f) =
such that
= !f@<iX
I@(f)
E:
H.
1\~;1Z(X)(R)~ is a dense subspace of
AZ;1 (X) (R). We should point out that the fact that the stochastic integral is
Z
not defined for every f in A2;1z(X)eR) is a consequence of :11e critical choice
of the constant in (3.7).
one and that
A~;Lz(X)(R)
We will see that the constant
(p+1)"2
is the logical
is large enough to include most integrands of interest.
For this we need to introduce the follmling notation.
all polynomials in the elenents of H(X).
Let
For each p 2: 0 let
P be the set of
Pp
be the set of
35'
e
all polynomials in P with degree no weater than p (PO
stants).
For each p<::l
~
let
is the set of con-
be the set of polynomials in
P which are
p
orthogonal to Pp-1 1 and let f20 = PO. The closure ~ of Pp in LZ(X) is
th hOrlogeneous chaos. 111e following a.re then clear or well known
called the p(e.g. lCa11ianpur
(1970)~
Neveu (1968) )
J.
P
=
P
(3.10)
t2q
~
LZ(X)
H@PeX)
for p;tq
pt!;
Q
q=O--q
=P =
$
p<::O
~
2 Oy,
and that the following is a CONS in each
~ ~
p<::l ~
A:
where
{~y~ y€r}
(3.11)
is a CONS in H(X).
AZ. L (X) (R)
~ 2
=
Lemma 3.1 and.(J;.·10) imply that
p
$
AZ.7j (R),
p<::O~:p
AZ' P (R)
1 p
111e basic properties of the stochastic integral
=
1
$
AZ' 0 (R).
q=O~-q
= ~cI o = ~o~*o~O-1 01 0
follow
from the following structure
-1
0
1
AZ;LZ(X)(R) ~ LZ(X)0H(X)
A~;L2(X)(R)
""
($ H0 P (X)) ~H(X)
p<::O
H*
=
-1
3.12)
~O
~0H(X)
e
~O
""
""
=
and are given in the following
""
H@P(X)0H(X)
onto
,
'"
~+1
36
TI-JEOPBil 3. Z.
The stoahastia integ'J:?aZ
1: AZ;L (X) (R) -+ L2 (X)
Z
denseZy defined aZosed Unear map with domain AZ;LZ(X)(R)
LZ(X)ePO ~ the set of all zero mean r.v.
e
fS
in 1Z(X).
is an unboundecZ
and range
L~(X) =
Henae every L2-funational
of X admits the representation
f
for some (non-unique)
and hence Az;p (R)
p
1 to A2 ; ~ (R)
f
€
H~fl(X)@H(X)
Since
AZ;~ (R) c
C
11*
~+1
with norm p+1.
If
l:p~o fp ~ fp € AZ;~ (RL then f € A~;LZ(X)(R) if and
< 00" and if f € A~;Lz-(X) (R) then I(f) = rp~ol (fp)'
=
that A2;~ (R)
p+l~
AZ;LZ(X) (R)3
and the restriation of the stoahastia integral
It is clear from (3.12) and the fact that for each
map with norm
from (3.1ZL
For eaah p~O~
is a bounded Unear map onto
rp~olll (fp)liZ
PROOF.
A~;L2(X)(R).
A~;L2(X)(R)~
AZ;L (X)(R) and f
2
only if
,..,
C
€
_C
AtLZ(X) (R).
Since
p~o~
Ifp is a botmded linear
so is the restriction of 1 to AZ;Op (R) and clearly, again
I(A2;~(R)) = ~;tl.
If* is onto
,..,
ep~lB@P(XL and qi($p~lffP(x))
= L2(x)e~ = LZ(x)ePO =
L~OQ s it follows that 1 maps its domain onto L~(X).
We now prove the claim in the last sentence of the theorem. Let
AZ;1 (X) (R) and write f = rp~Ofp ~ fp € A2;S? (R) • Then f € Ai; L (X) (R)
2
2
is equivalent to qioloI@(f) € H~s and since qiol.r (f) = rp~o~~lOI@(fp) and
f
€
each Ci?Olo!@(fp) belongs to H@P(X)@H(X)s by the definition of 11* this is in
turn equivalent to
r IIIf e4.i lo !@(f)11 2 < 00.
p~o
But now I!lfpoqiolor@(fp)11
p
o
P
= I Iqio'Yp.qiOlo!@(fp)11 = III(fp)ll. It follows that
37
f
E:
f
E:
A~ ;L (X) (R)
if and only if Lp;,::O 111 (fp ) II <
AZ;L (X) (R)
2
it £0110",s from the definition of If*
2
= ~Olf*o~~l.I~(f) = r
ref)
p;,::o
00.
~0o/ o~~loI@(f)
Assuming now that
that
= r
p;,::o
pp
r(f) .
p
In order to complete the proof of the theorem we need to show that
closed.
A~ ;L (X) (R) = V(1), f n -+ f in
2
We will show that f E: Vcr) and l(f)
in LZ(X).
Since
1 (fn )
~
Let
fn
E:
E:
Also
since
1 (fn )
fn
-+
f
-+
e,
Write
r
1 (f) = Lp ;'::OI (fp )
by Fatou's lemma,
and writing e = Lp ;,::oep+l
2
~
= e,
is bounded we
By Fatou 9 s lemma we have
L 111(fp)-ep+l ' 1 = p;,::O
L
p~O
showing that 1(f)
restricted to AZ;~ (R)
sho\tJing (by the last claim of the theorem) that
Thus
lim
n
liml 11(fn p)-ep+ll
n
'
:1
ep
E:
f
V(l).
€
~, we have, again
2
1
r 'IZ(fn ,p)-e
+1 112 = liml 11(fn)-el' = 0
p
n
p<::O
0
which concludes the proof.
The same argument can be applied to define spaces
A~;L2(X)(R)
-+
implies that for each fixed p;'::O,
in AZ;~ (R), and since
r(fn,p) n+. r(fp ) •
have
is
V(I), by the last claim of the theorem (just shown) we have
= rp~OI (fn,p).
fn,p ~ f p
r (~)
AZ;L (X) (R), and
Z
= e.
r
and the stochastic integral
J(f)
A2 ; L (X) (R) ,
= ff(t)Xtdt
Z
for
e
38'
f E A~ ;L (X) (R), ",here we assume accordin.gly that X is a zero mean mean
2
square continuous Gaussian process. It can also be shown that A~;L2(X)(R)
A~;L2(Z)(r)
and ff(t)Xtdt
= ff(t)dZ t
=
where r and Z are related to R
and X as in Section 1.1.
We now consider some of the properties of the stochastic integral.
First
we show that It6's integral is a special case of the general stochastic integral defined here. The proof is based on (i) of the following lemma which
will be also useful later.
LErj~
3.3.
= an.
(ii) If
(iJ
If a E L2(X)
n e H(X)
and
are independent then
Il'(a@n)
a,n E H(X) then Il'(S@n)
- E(Sn).
(i) Assume without loss of generality that E(n 2)
PROOF.
{n, l;y' YErL
r Ii11early ordered, by a CONS in H(X).
representation of S e L2(X)
a=
= an
= I~
By the Cameron-Martin
there is a countable subset r'
of r such that
Po,Pp
,Pk
ay
Y
hp Cn)Hp (l;y )
r
L
r
1"'" k
0
1 1
p~O PO+Pl+· .. +Pk=P Yl""?Yk Er '
eo.
k~l
and let
T
••
••. fL (l;y ) .
Pk k
Yl<oo.<Yk
Thus e is a function of the r.v.'s {n? l;y' YEr'}, and since n is independent of the r v. 's {a} l;y' YEr'} it follows by an elementary property of
0
conditional e>qJectations that a is a function of the rov.'s {l;y' yer V}
\ and in fact
a
=
E(e/l;y' yer').
It then follows from the series expansion of
e and E(~o(n)/l;y' YEr v) = E(f\JoCn)) = 0, PO~l,.that
a
O,Pp'·· 9Pk
Yl'···'Yk
=
L e
p~O
p
only
H (l; )
Pl Y1
00'
H (l;y)
Pk k
39
11~O.
.
We first show that 80n
E
and we h;w€:.;8
Since <iI-l(El)
P
=
€ 7'j.
0
-P'
-
l:p~oq,-l(apL
H or equivalently that ~;l(e@n)
4l- 1c ap )
= qi-l(8)@n
E:
n0P cxL this is ecpivalent to
E:
I l~p(~-l(eD)@n) I 1 <
2
l:
p~O
00.
I.
It follows from the expression of each ap given in (3.13) that
q,
-1
(8 u)
..
~
= (pI) l:
a
- @Pk
O,Pl,···,Pk @Pl-
Yp".'Yk
~
Yl
0 •.• @~y
··
k
and, by C3.7), that
~)(<p-
(3.14)
111US
1
~
~
O,Pl,···,Pk 0Pl_ - @p~
~
@••• 0~
@n.
YI'···'Yk
Yl
Yk
C8p)0n) = {(p+l)!}2 L a
we have
1
lI~n. (~- (8p)0n)
~
2
II = (p+1)!
O,pp .. ' ,Pk) 2 PI!.· .Pk!
r (aYp
..• 'Yk
(1'+1)!
= I Iqi-l(8p) I 12 = IISpl12
ffild hence
a0n
E:
rp~ol l~p(<p-l(8p)@n)
2
I1
= l:p~ol lapl 12 = I lal 12
<
00.
H.
I.Jow from the definitions of
~(80n)
~,
(3.9), and
~*
= q,o~*e<pol(a0n) = q,o~*(~-1(e)0n)
= q>( l:
p~O
~p[q>-l(a )0nJ)
p
= L 4l(~p[q,-1(8D)0n]) .
p~O
For each
P~09
we have
using (3.14) we obtain
..
It follows that
H*.
40
= 8p n
~~d thus
~(e@n)
= lp~O
en
= en
r.:>
(ii) Now let 8~n
e
= E(8n)n
t(8n)n@2
+
+ ~
H(X)~ assun~ again tl1at ECn 2)
e =8 -
~rrlere
~@n and
E
by
as desired.
=
1~ and ~rrite
E(8n)n is independent of n. Then 8@n =
Ci)~ ~C8@n)
= E(~n)~Cn@2)
+
~n. But ~(n@Z)
=
¢o~*o<P(/Cn@n) = ~o~l (n@n) = 12 <p(n*n) = Hz(n) = n2-L Hence
o
THEOREt:'! 3.4.
ItaYs integ:mZ for the fliener process is a speaiaZ aase of the
stochastic integ:mZ
PROOF.
r.
Let X be the Wiener process ~ i. e.
integral, denoted by
r*,
R(t ~ s)
Then It6 vs
defines an isonorphism froD nZ onto
is the Hilbert sabspace of L2(S1xR~ i3(X)xBCR), dPxdt)
all elem.ents ac.apted to
= tAS.
X.
Note that
i1
2
c:
= LZCdPxdt)
L~ (X), where [1 2
consisting of
L" (dPxdt) = L2 -L ('.r) Cdt) =
~
, 2A
H
A2 ; L Oq(R). Let ~,1 be the set of all elements of the form Iifnl(an,bnJ,·w~lere
z
1~ L c',r)
.
B"'u'
(Y
)
, 1
[\,1'
~
-n ~ 2.l~ and.l:.en J.S
'L~an -measuraD e.
d
J.S a nense
subspace or,. i:~ 12
and that 1=1* on M.
1*(f)
Since each f n
by Lemrna 3.3(i)
= £.1
~Nfn C'r.t\b _v
)
.t\.a
n
n
•
41
that IfeD<=
L~fn0(\
H~
-X ) e
n an
and thus
I (f)
fer 1Z • TIlen for some
J
f.1
c L~ ;1
Z
f n e r,~
c L~ ;L
properties of Itep s integral that
sho~m9
that
1*(fn) = 1(~),we have
f
€
Z
and
= L~fn (Xbn -xan).
= 1* (f) •
,Finally we show that
.1,\
L~'L (X)(dt)~
?
l(f) = ~(rf0dX)
Hence
fe
L~;L2(X)(dt)
L~;LZ(X)(dt) = AZ;L (X)'(R)
2
and that
1=1* on r'I Z '
Let
(X) (dt) ~ f n -+ f. It follows from the
Z
1*(fn ) -+ 1*(f) and since 9 as it was just
1(fn )
1(f)
and
(X) (dt)
2
-+
1*(£).
= 1*(£).
and thus
Since 1 is closed it follows
Clearly NZ is a smaller class than
1 provides an extension of 1*.
0
We now consider the problem of calculating the stochastic integral for
specific integrands 9 starting with the simplest possible case where f(t)
with 8 e LZ(X)
= 8$(t)
and $ e AZ(R).
THEOPJ3H 3.5.
(i)
If 8 e
L20c)~
ep
IE
AZ(RL and
e
and 1<P(t)dXt
ewe
independent then
(ii)
If
ee
H(X)
and
fe$(t)dXt =
<P
e AZ(R)
then
e f<P(t)dXt - f(e f<p(t)~) •
2
If for some ueT~ F(x) IE LZ(R, exp(-x /ZR(U,U))dx) and if F(x)
L2 (R, exp(-x2/ZR(U,U))dx)-derivative denoted by P (x), then
(iii)
has an
fF(XU)$(t)~ = F(Xu)J<P(t)dXt
- FY(Xu)E(Xuf<p(t)dXt ) .
42
PROOF.
We first show that if 8
Indeed, if cj>
s~~le
is a
function
€
cj> =
I@(8cj»=I@(2rIJ8Cleb]) =
nan' n
and since
r0
LZ(X)
and
cj> e A (R)
2
2~Cnl(a b]
we have
n' n
2~(8C
n
then
bn -Xa n )=8@fcj>dX,
)@(X
is an isomorphism the same is valid for all
cj>
E:
AZ(R) •
It follows
that
(i)
Let
l(f) = eIcj>dJc.
1tVith each
= 8cj>(t).
f(t)
"t'lrite
f e ~;L2(X)(R)
We will show that
e = 2p~08p'
fp e ; .
Then
f =
2p~08pcj>
in L2;~ (R) 0 VIe first calculate
fp(t) = epcj>(t)
= V(l)
and
in AZ;L (X)(R)
2
1 (8 p cj> (t)) 0 Note
that it is clear from the proof of Lemma 303(i) that the independence of 8
fcj>dX
(=
neH(:)<))
implies the independence of each
and fcj>dX.
8p
and
Thus by
Lema 3. 3 (i)
1(8p cj»
= ~oI@(8pcj» = ~(8p@fcj>dX) = 8p fcj>dX
Now the independence of 8
p
and
fcj>dX
2
= L
•
II 1 (8p cj» II = 11 8p ll
implies
Q
IIfcj>dXll
and thus
L II 1(f:p) 11 2 = l:
p~O
p~O
111(8 cj»11
p~O
P
1I0pl12l1fcj>dXlIz
f e Vel)
It follows from Theorem 302 that
and that
l:p~o 8p fcj>dX, and again by independence we have
(ii)
If
e
€
1(8cj»
H(X)
and
cj>
€
AZ(R)
then
= lIel12ollf¢dXllz
1 (f)
= l:p~o
1 (fJ
<
00.
=
1 (f) = 8fcj>dX.
eep
= ~oI@(8cj» = ~(8@fcj>~~) = ef¢dX
€
A2 .() (R), and by Lemma 3.3(ii)
~~-l
- ECefcj>dX) 0
43
(iii)
First let
=H
FCXu)
2C~),
p,cr
u
letting
J¢dX
=n
and noting that
p~l~
where
<1>-1 (H.. 2 (~) )
p,cr
u
JH
p,cr
2(Xu )¢(t)dXt
2
cru
= E(Xu2) = RCu,u).
= (p 1) ~~P
= ~eW*((Pl)~X~P~n)
Then
1Ale have
.
u
Write
n
-Z
= cru
E(XUn)~ + l,;
where
l,;
=n
-2
- cru E(~n)~
= Cp+l)~{cr-2E(X
n)x@P+l
u
u
u
t/J ('C0Pqp n)
P 'u
J.
Xu :. Then
+ X0P@Z;;}
-"U
and thus
Jffp,cr2(Y'u)<t>Ct)d~ ccr~ZE(Xun)H p+l,cr2(~)
u
u
=H
p,cr
+ H z(\t)l,;
p,cr
u
2(Y\1)n + cr-ZE(xunH H
ZO~) u
p+l,cr
u
=H
p,a
2 (Xu)}
u
zCXu)n - pECXun)H
Z(Xu)
p-l,cr
u
\llhich is the desired relationship since
F(Xu )
p,cr
u
u
Now if F (~}
~H
dH
Z(x)/dx = pH
Z(x).
p,cr
p-l,a
is as in the statement of the theorem we have
= 1:
p~O
apH.
.2 (Xu), F ' (\t)
p,a
u
= l:
p~l
papH
Z(~)
p-1,cr
Since for each p~O,
with both series converging in LZ(X),
u
II I (H
Z(~)¢)
p,cr
u
AI
1:
p~O
Ill(apH zO\)¢) liZ::;
p,cr
u
L (p+l)a~IIH
z(Xu )II Zo ll<t>II
p,cr
u
p~O
= (II F (~) liZ
i
It follows
by Theorem 3.2 that F(Xu )¢
€
+
Z
II F (~) liZ) II ¢ 11 2
Vcr) and
<
00
•
II ::;
44
fF(\t)¢(t)dXt = r fa H
2(X )¢(t)dt
p P~O"u u
p;;::O
=
r
{anH..
p;;::O
>:
2
p$O"u
= F (Y~) JC/>dX
O~) f¢dX
- Fi
- papH
2 (X ) EeXuJ¢dX) }
- p-l~O"u u
(~JE O'uf epdX)
0
•
H 2(~)' and exponentials,
P'O"u
exp(~-~O"~) $ and it admits a natural generalization to Ps of the fom
(iii) includes the cases of Hermite
As an illustration we lITite the following simple integral
F(Y\! , ••• ,Xu ).
1
polynomials~
k
J~Y\r¢(t)cL\ = V\rfC/>c1'{ - E(~f¢dX)~ - E(\rf¢dX)~
•
Before evaluating some less trivial stochastic integrals we consider the
following interesting result.
e
Let T = [a,b] and a = to,n<t1,n< ••• <tn,n = b$
n=1,2, ••. , be a sequence of partitions of T whose mesh goes to zero,
maxi (ti,n-ti-l,n) .... O.
The mean square quadratic variation of X on T along
such a sequence of partitions is defined as the mean square limit of
2
I~=1 eXt. - Y't.
) whenever the later exists.
l,n
THEOREM
I-I,n
3.6.
Let X =
{J<t, t€[a,b]}
be a zero mean Gaussian process with
continuous covariance R of bounded variation on
measure on [a,b]x[a,b]
[a,b]x[a$b]
corresponding to R is denoted again by R).
mean square quadratic variation ~ of X on
D~ is the diagonaZ of [a,b]x[a,b].
Then the
[a,b] aZong any sequence of
partitions whose mesh goes to zero exists and is given by
where
(the signed
45
PROOF.
we have
!rl<xt~xs>ldZIRI(t~s)<oo
(a,S]c[a~b]
1.1 that
By the mean square continuity of X and the bounded variation of R
and 8t=:L Z(X).
~ t=:
and !fl<xt,el(a,S](s»ldZIRI(t,s)<oo for all
It then follm'ls from the extended version of Theorem
AZ;H(X) (R) c V(l)
alld thus the stochastic integral
fXtdXt
is
defined.
Let a = to,n<tl,n< •.. <th~n = b be any sequence of partitions with mesh
n
n
) is defined by xi ) = xt.
on each (ti_pt i ], then
tends to zero. If
xi
X(n)
1-1
X in AZ;H(X) (R) (by the mean square continuity of X and the bounded
variation of R) and hence J'Ain)dXt -+ fXt~ in L2 (X). Thus, by Theorem
-+
3. 5 (ii),
Subtracting (3.15) from (3.15 V) gives
n
o = lim {L
n
i=l
eX
ti
-x
t i -l
2
n
) - r
i=l
E[(Xt -Xt
i
2
)]}
i-I
and since the second term has limit R(D~), the result follows.
o
Notice that by adding (3.15) and (3.15') we obtain
b
f
a XtdJ\
where cr~
=:
E(~)
= R(t, t).
2
= !a(Xb
Z
Z
Z
- Xa - °b + cra )
A similar approach leads to the following result.
45
THEOREM 3. 7.
Let
X be as in fl"heozoem :5 6.
:fb
.
H
,
2(X t )dXt
ap,ot
f: explXt-""~Jdl\
PROOF.
AZ; ~ (R)
0
Then
0
1{H
= p+l
=
}
2 (Xa ) ,
2(Xb) - H
p+l~ob
p+l.oa
p~O
expC4, -""~J - expCx,. -""~J .
It is shown as in the proof of Theorem 3.6 that H 2 (~) €
p,Ot
Letting a = to,n <t1 ,n<. <tn ,n = b be any refining sequence of
0
•
partitions with mesh going to zero, and using the (tmifonn) mean square continuity
of X and the bounded variation of R, we have (writing t i
simplicity)
fb H
2CXeJdXe
a PSot
ti,n for
= ~.~··~ol(fb H 2CXeJ~1
a p,Ot
= ~.~ o~Ol{lim ~
n
P
r
for
= {(p+l)!}~
1
H 2
(~. )~(Xt.-Xt. )}
p,o
·1-1
1
1-1
t.1- 1
<li{lim
n
r. x@Pt. 1;(X
1
!
= {(p+l)!}~ \Il{lim L 1+
n i P 1 m=O
= {(p+l) !}~ ~{lim l
p+1
~
= {(p+l)!}2
p+1
1.,
= {(p+l)!}~
p+l
!
t.1- 1
1
)}
x';m ;xtip-~(X -)7
)}
t i -1 t i
t i \i-l
(X@m
n 1. m= 0
-X
"L.
1-
@;p+l-m _ xem+l@}';p-m)}
t.1-1 ~.1
t.1- 1 t 1.
,..",
ttY
\Il{lim L (xt8p+l _ ~P+l)}
n i
i
i-I
,.."
~(X~+l - XSP+l)
0
a
=__1__ {H
The second result follows from the first and
p+l
(X )-H
eX )}
P+l,o~ b P+l,O; a
47
exp(~-~(J~) = I ~ H
P~O
Putting f (t)
P
=~ H
p.
I II I (f
p~O
fa
Z(JC ) . it is easily seen that
P,(Jt t
~
) 1/2
P
Thus by Theorem 3.2~
b
z(Xt ) .
P~(Jt
2
Z{! 'I exp(X -~(J~) liZ + "exp(X -lz(JZ) 11 } <
a
b
a
exp(~-~(J~)
2
exp(\-!:2(Jt)dXt =
:=
€
Vcr)
1 Jb
I pr
•
and
2(~)dXt
H
p~O
a
I
(p+b! {H
p~O
00
P~(Jt
z(Xb) - H
z(Xa )}
p+19(Ja
p+l~(Jb
o
Theorem 3.7 shows that Hemite polynorrlals H 2(Xt ) play the role of
P,(Jt
customary powers,
and exp (Xt -~(J~) the role of the customary exponential ~
xi,
~
exp(Xt ), in this stochastic calculus.
48
Throughout this section we assume that X is as in Theorem 3.6. We first
e:h'P1ore the connection between the IVJkH i 5 and the stochastic integral. We want to
establish that each MWI can be written as an iterated
f
P
integral~
Le. ~ that for
e: AZ(iPE) ~
(4.1)
IT ...
where of course the iterated integral remains to be defined.
Let H be a Hilbert space and
f(t l , .•. ~ t p )
on TP.
Then sf~~
sf~~
, the set of all H-valued step functions
is an inner product space with i.."Uler product
and its completion is denoted by lI.Z;H Cel'R). It is easily seen that AZ;H(0PR);;;:
AZC~R) 0H under the correspondance (<1>1 0 ..• 0<1>p) S ~ C<1>1 0 .•• @<1>p)@S . TI1US each
element in A2;r/~R) hB.s an orthogonal development of the form
I
where
{<1>y' ye:r}
aa
Yl~···?Yp
and {s~, ae:A}
(<1> 0... @<1> )S
Y1
Yp a
are CONS9 S in AZ(R)
and H respectively.
Consider the following c1lE.in of maps
1T
P-l
A
~~~H
defined first by
1T .
( ••.• ) . ; J
~
·K
2;~_1 ~
Q..
P
49
Then~
by the same argument used for defining 'lip] each
bmmded linear onto Cnot one to one)
with norm
IIk1.p
cr.
1
1T
q
can be extended to a
It is important to note
p is the stochastic integral~ and that on ~_l-valued step functions
acts like stochastic integral by fixing the l1 extraH variables. The iterated
that
1T
integral in (4.1) is now defined to be
Letting T
=:
1T
q
l Cfn"' ) and the equation follows.
a.71d noting that IpCfp) =: IpCfpL we should expect to
[a~b]
1T •••• o7f
p
obtain from C4.1)
b
I fp(tl~···»tn)dXt
a
"'
1
••. dXt =:
P
.
=:
p!
I lCJ
b
a
t
t
p ••.
a
(Jr
a
2f
P
(tl, .•. »tp_l»tp)dX~ ) ... dXt
~
b
=:
J hp(tp)dXt
a
ma~e
)dXt
A step function
r~ fnlCan»bn] in A2;L (X)(R) is called
B(:t~
adapted if each f n is
a~t~an)-measurable.Theclosed subspace of A ;L CX)(R) generated by the
Z
2
2
ad
adapted simple functions is denoted by AZ;LZ(X)(R)
and its elements are called
adapted. We also let
ad*
ad
(
112 ;L (X) (R) = 112 ; L eX) R) n A~; L (X) (R) •
2
2
2
LBlfj\1A 4.1.
t
·t
P( •••
fo
P
precise (in the proof of
Theorem 4.2). The following definition will be used.
=:
p-l
p
where hp is adapted to X. This will now be
f
~l
t
If
f
€
A (~R)
2
is a step funotion then
(Ja2fCtl~···~t P_l,tp )dXt 1) ••. )illCt p-l
g(tp )
=:
.
is an adapted step funotion and
50
where Doth integraZs are ckfined in the usuaZ way as the oorresponding integrals
PROOF.
For ease of exposition we only consider the case
p>2 being similar.
the case of
It is then sufficient to prove the assertions for f of the
fOTIll
(i) 1(a~B](tl)l(Y1o](t2),· (ii). ~(~~Sl(tl)~(a~(3](t2L
(iii)
1(y~oJ(tl)1(a~(3](t2)
(i) (Xs-Xa )1(y,o](t 2L
step ftmction.
p=2~
where a<(3<y<o. Then g(t Z) equals
(ii) (Xt·2-Xa)l(a,I3](t2)~ (iii) 0 and is ,thUS an adapted
Using Theorems 3.S(ii) and 3.7 we find that the right hand side
of (4.2) equals
b
(i)'
fa (X{3-Xa)1(y~o](t2)d~2
= (XI3-Xa)(Xo-Xy)-E[(Xs-Xa)(Xo-Xy)]
= 12 ~{(X(3-Xa);(Xo-Xy)}
(ii)
fb(Xt -x )l(
a
2 a
Q](t2)~~t2 = fl3aXt 2dXt 2 - f(3x
~~t
a a 2
a~~
= ~{X~-x~-cr~+cr~}-{Xa(XS-Xa)-E[Xa(XI3-Xa)]}
=
(iii)
~{(XS-Xa)2 - E[(XS-Xa )2]}
fba OdXt2 =0.
On the other hand the left hand side of (4.2) equals
(i)
I2(1(a~S](tl)1(y,o](t2)l(tl<t2))= 12 ~{(XS-Xa);(Xo-\)}
= (Xs-Xa)(Xo-Xy)-E[(XI3-Xa)(Xo-XY)]
51
o
and the proof of the 1errnna is complete.
THEOREM 4.2.
I(1\.~~~2(X)CR)) = L~(X)
e
and thus eaah L2 funational
of X
admits the stochastia integral peppesentation
e-
fee)
whepe (the not neaessaPily unique)
e
PROOF.
first that
fn
€
f
=f
f(t)dXt
is adapted
It suffices to prove the second assertion of the theorem.
8€~ so that by Theorem 2.1~
e = IpCfp ) = Ipcfp)
Assume
for some
1\.2(~R).
"-
If
<p
is a step function in 1\.2 (J'R)
it is easily checked that
and
where n
= Cn1~ •.• ~np)
such permutations.
with Cpn
-+-
fp •
is a permutation of
Now let
Then
{<P }
n
(l, •.• ,p)
ffi1d
TI
is the set of all
be a sequence of step functions in 1\.2 (oJ'R)
52
e
implies that {¢nlCtl< ... <t )} is Cauchy and we denote its limit by
p
fplCtl<.oo<tp)' Then
IpCfp)
= l~
If we let ~1
IpC¢n)
=
l~m pl Ip(¢nl(tl< ... <t
n
= Wp-lo •..•Wl(Pl¢nl(tl< •.• <tn))$
tion in AZ ' 0
113- 1
It follmvs that
adapted to
(R)
~
+
h
E
p
For a general
)) =
then ~ is clearly a step func-
by Lerrm.la 4.1 ,
X
pl Ip(fplCtl~"'$tp)) •
and
by the continuity of
7f
q
1s
Wp_loo' ••Wllul f p 1(t <••. <t )) = hp •
n
1
AZ ' 0 (R)
1--p-l
is adapted to
X
and
e E LZ(X) we have by Theorem 2.1
e - f(B)
=
L
p~l
Ip(fn)
=
1
satisfies
and the above
L I(hp ) = I( L h)
p~l
p~l
P
belongs to A~9L2CX)CR) (by Theorem 3.2 1 since
Lp~llllO~) 11 2 = Lp:'d II Ip Cfp) 11 2 < 00 ) and is also adapted to X since each
his.
0
where h
= Lp~l ~
P
It is clear from the definition of fp(tl$ ... ,tp)letl< .•. <tp)
in the proof
of Theorem 4.2 and from Lemma 4.1 that equality (4.2) is valid for all
fp
E
AZ(@PR) where both integrals in (4. 2) are defined as the corresponding
integrals of fp(t11 .•. ,tp)lCtl< •.• <tn) •
>;
We finally consider the stochastic integral of nonanticipatory functions.
A
step function f = L~ f nl(a b] in AZ' L eX)(R) is called nonantiaipato~ if
n' n
Z
each f n is independent of the increments of X after an' The closed sub$
space of AZ;LZCX)(R) which is generated by the nonanticipatory step functions is
53
e
denoted by A~L (X) (R)
and its elements are called nonanticipatory.
2
THEOREM 4.3. J\.~~L2(X)(R)
Az'L (X) (R).
?
2
A~;12(X)(R) and the stochastic integraZ
A~~L2(X)(R) is no~ preserving.
restriated to
PROOF.
c
r~f 1 (
b 1 be a nonanticipatory step ftmction in
n an~ n-Since for each n, f n is independent of x. -x
,it follmvs
Let f =
--~
~
from Theorer.l 3.5(i) that
Then
N
2
111(f)1 1 =
I
n,m=l
E{f f.(Xb -x )(Xb -x )}
n m n tl:n
m. am
E{f2 (X'. -X )2} = E(f2)E{(X, -X )2}. For ~ < am we have
n nbn an
n
D an
n
from the inde-pendence of f
and f
from X -x
that
n
m
b.m amI
For n=m we have
It fo110\'1s that
Ill(f)11
2
co
E(fn fm)E[(Xb -Xa )(~ -Xa )]
n n
m Iii
2
<f(t),f(s» dZR(t,s) = IIfI1
= L_
n,m-l
= If
and thus the stochastic integral is nann preserving for step nonanticipatory
S4
e
ftmctions.
Now let
II I (f) II
==
functions
f
E:
II fll.
f
A~~LZ(X) (R)q We will show that f
By definition there is a sequence of step nonanticipatory
n such that
L
f =
f
f
f
,
L
p~O
~
n
..
2
=
r
II fn-fmII
f
A~~L2(X)(R)
c
f
fp,fn n
;
n,p
r
II £n p liZ
p~O'
A~ ;1 (X) (R) •
2
€
AZ"-Op(R) •
~..,
2
~L
implies that the sequence
have 1 (f) = lim 1 (fn )
nence
€
.
n
= liJn
Hence by Theorem 3. Z,
r
D~O
lin I 11 (£n -0)11
p~O
n
=
n
Write
1 (£n p) ~ 1 (f ) •
p
,
n
f n and thus
I II(fp )11
f.
-+-
n
. p~O P ~
Then £n p
A~;LZ(X) (R) and
€
It follows by Fatou is lemma that
s Ilia
n
= lim
n
Now
L
p~O
II f n liZ
=
n
na n
Z
<
00
•
111 (fn -fro) II
1(fn ) converges ~ and since
A~;LZ(X)(R) and 1
mOl.., how large
I(A~~LZ(X)(R)) = L~(X) or
II £ liZ
Ill(f)11 = 1imI11(fn)11 = lim
&,d thus
I
Ilfnll
restricted to AZ;LZ(X)(R)
=
is closed we
= Ilfll·
is a norm
o
1 (A~L2 (X) (R))
interest to
=
111 (fn) - I (fro) II
preserving map into L~ (X) •
Note that
I 11(fn p)II
'
is a closed subspace of
L~ (X) and it would
be of
it is in general, and under what conditions we have
equivalently that each LZ-ftmctionalof X has a
nOltanticipatory stochastic integral representation
8 - fee)
where f
is nonanticipatory.
= J f(t)dXt
We conj ecture that this would be the case if the
germ a-fields of X are trivial.
e
The notions of iladaptedli and of "nonanticipatoryYl introduced above are of
course identical
~rllen
X is Wiener process.
55
5.
NDr-JLWEAR NOISE
A (strictly) stationary process Y = (Yt~ -oo<t<oo)
is called noise.
e(~t
YO =
e:
out; the noise
(Yt ~ -oo<t<oo)
input by the flow of the white noise rJ(.)
e
e = or f... f
p~l
fp
€
L (RP), the noise
2
through the nonlinear device
(5.2)
~~( o+t).
In order for Y to be a
has the orthogonal development (cf. Section 2)
(5.1)
where
-+
is produced by shifting the
E8=O and Ee 2<oo.
noise we require that
Since
You put in a white noise
(fonna1ly the derivative of a Wiener process tV) and you get
-oo<t<oo)
t
EY~<oo
Quoting from McKean (1973), Wiener (1958) liked to think of such
a noise as the output of a i1black box"
fJ = (t?Jt ~ -oo<t<oo)
with EYt=O and
Yt
f (t 1 ,···,tu ) dWt .•. dWt
P
~
1
P
Y obtained by shifting the incoming white noise
e
can be expressed as
= l f···f fp(tl-t~ •.• ,tp-t)
p~l
dWt ... dWt '
1
P
and the covariance function of Y is readily seen to be
EY Y = I. pi
t s
p2:l-"
f···f 1
P
~
(L = t-s)
(t1,···,tp)f (tl-L, •.. ~t -T)dt1···dtp •
P
P
Wiener's theory of nonlinear noise starts from this idea.
He also proved a
profound theorem which was clarified by Nisio (1960) and which states that every
ergodic noise can be approximated in law by noises of the form (5.2).
not every ergodic noise has the representation
{5~2L
Note that
and a necessary condition is
strong mixing (lVlcKeail (197:))). (For a discussion of the ergodicity of stationary
processes and processes with stationary increments see Doob (1953).)
56
e
Mbre generally~ instead of sending white noise (or Wiener process) through a
nonlinear device 8 ~ we may send a Gaussian process with stationary increments
X = (Xt~ -oo<t<oo) ~ with say
a.s. and covariance R.
XO=O
Then the noise Y
obtaii1.ed by shifting the incoming Gaussian noise X can be expressed as
(5.3)
where
f p e: AZ(J'R) ~ and the covariance function of Y is again readily seen to
by
(S.4)EYt Ys c P!lPIJJ... JJ
=
where
e
T
= t-s
I
p~l
pI
<£
~P(tl.···,tp)fp(Sl-T•.•.•Sp-T)d2R(tl·Sl)···d2R(tp'Sp)
f (o-T~ ••• ~e-T»
(o, ... ,oL
P
P
.
A2(~R)
(cf. Tlillorem 2.1).
clear~
Although (5.2) and (5.3) are intuitively
they require proof.
The
proof of (5.3) follows from the followL"1g property.
LEI',t1A. 5.1.
If X is a zero mean Gaussian process with stationary increments
and covarianae R then 3 for
fp
€
A2(~R) ~
J... Jfp(tl,· •• ,tp)dXtl+t···~p+t
= J••. J£P(tl-t' •..• tp-t)~l···dXtp .
Both integrals are well-defined since X has stationary increments.
....
....
~Pl"""
.... @Pk
Pick a CONS {</>y~ ye:r} in A'Z(R). Since {</> 0 ••• @rj>:YlJ
~YkE:r,'
"(1
Yk
Pl+. "+Pk=P~ k~O} is complete ~ AZ(0PP~ and Ip(fp) = IpcfpL it suffices to
PROOF.
GO
prove this assertion for
becomes
f
P
= </>
@Pl.....
Y1
..... 0Pk
0 ••• ~</>
Yk
.
But for such f p
•
~
the assertion
57
Jep(u)dJ\x+t = I
This is true for
<peSI
and hence for
<P
(uet)~
<PeAZ(R).
.
The proof is now complete.
itJhen Y has representation (5.3) $ we say that Y is X-presentable.
that X is always X-presentable since ;~ = !~ dYu '
THEORE~
stationaPy
5.2.
sho~ln
The
for X-presentable processes.
Let X be a mean square continuous Gaussian ppoaess with
incpements~
Xo=O
a.s.~
and
~ith
absoZuteZy continuous speatpaZ dis-
Then every X-ppesentabZe noise Y is stpongZy mixing.
tpibution.
Having introduced the Fourier transfonn on AZ(~R)
the proof is identical to IiIcKc:;..n i s proof for X Wiener process.
in Section 1. 3 ~
PROOF.
that if
Note
As McKean (1973) showed~
if Y is not strongly mixing then Y is not l'Jiener process-presentable.
same property is now
0
nR
We need to show
is the set of all real valued functions defined on R~ and B(RR)
the a-field generated by the cylinder sets of
lim Pr(YeA~ y1'€B)
nR ~
then for A,B
€
is
R
BCR )
= Pr(YeA)Pr(YeB)
1'400
\~ere
y1'
denotes the shift of Y by 1'~ i.e.
lim E(ele~)
1'-+00
where
e l'
In fact we will show
= E(8 l )E(6 Z)
denotes the fimctional of shifted paths ~ i. e.
6l' (Y)
= e (Y1') •
Then the
58
e
strong mixi.l1g property is self-evident.
Since Y is X-presentable,
LZ(X), and we can expand 81' e2 as follo1rls
ei
where
fi,p
c;:
= E6 i
+
AZC&PR).
E(e1e~)
u
I
p~
f···f
f i p(t1,···,t )dXt •.. dXt
~
P
1
P
We then have
= E(8 l )E(6 Z)
+
r
p~l
pI
<f1 , fT >
,p
1.10,
<£1
(where
T
_ A
AT
I
-
Z,p AZ(~PR)
(T , .•. , T) • By Theore:ms 1.8 and
J I iT(Al+"'+~)
'f
>
- <f1 ' f Z >
- . •• e
~P z,p A (oJ'R)
,p
~P L (IP)
Z
Z
F
is the spectral distribution of X), which approaches
the Riemann-Lebesgue theorem.
p~l
c
(i=1,2)
f~ denotes the translation of f p by
where
LZ(Y)
pI l<f1 ,f~ >1 ~
,p
,p
I
p~l
~~
0
T~
as
by
Also
pI
r
p~l
II f l ~ pI! II£z , pI!
P!(lIfl
~1I2+llfz p112)
n
,-
=
~(Var6l+Vare2)
where for each p~l, noms and inner products are in A (~PR) •
Z
<
00
It then follows
o
We now show the· analogue of Wiener-Nisio v 5 theorem using Nisio vs approach as
simplified (for convergence in law) by McKean (1973).
X will be a zero mean
sample continuous ergodic Gaussian process with stationary increments which
satisfies XO=O a.s. and the following condition
59
(S)
THEOREM 5.3.
Every measupab'le ergodic noise Y (de fined on any ppobabiU ty
space) is the timit in 'law of a sequence of
PROOF.
~ppesentab'le
Examining iocKean Ys (1973) proof for
noises.
X the Wiener process \'/e see
that the argument remains valid for the present general case if it can be shown
that there exists a sequence of nonnegative functionals
Bn
such that the probability distribution of each
its density ftmction is constant on
to construct such an's.
process, i. e.
{t:
(Q,B,P)
[O)n]
is absolutely continuous and
and decreasing on
For simplicity we suppose that
= (RR ~
B(RR),
an on the paths of X,
(n~co).
We proceed
X is a coordinate
p) and }~ (w) = w(t). Consider Sew) =
t.~.w>O}~
w€RR. Because of tile continuity of the paths of X, Sew) is an
....
open set a.s. and can therefore be expressed as a denUI!lerable disjoint mion of
open intervals
Ii (wL
Sn(w)
i~l.
Let
= ~{Ii(w):
IIi(w)l>n, Ii(w)c(-n,co)} .
1
Now we show that
Sn(w)
is nonempty a.s.
Let
depending on w only through its increments.
have
J
a+1'
. 1
11m
~
f(wt)dt
1'-+00
a
few)
be an integrable r.v.
Then by the ergodicityof X we
= Ef
a.s.
60
e
and a=-n.
Then it follows that Sn(w)
is nonempty a.s. if £f>O.
Ef = Pr[~tX>O, O~t<n; A_nX<O] > 0 by the assumption
(8).
nonempty a. s.
Note that
We nmV' define an (w) =
11
+ inf $n ((J.l).
But indeed
Thus Sn(w)
O~a
n
is
<00 a.s.
The same argument as on p. 211 of Nisio (1960) shows that an has the desired
probability distribution.
o
Thus the proof is complete.
We now give a discussion of assumption
(8) .
We believe that
(8)
always
holds when X is a zero mean sample continuous ergodic Gaussian process with
stationary increments, yet we are not able to prove it.
following sufficient condition for
Instead, l'Je have the
(5), which indicates that
(8)
is a mild
assunrption (if it is a restriction at all),
For each n~l there is an fn€R(C), the reproducing
kernel Hilbert space of the
covarim1ce G of X,
such that Atfn>O for t€[Osn] and A_nfn<O.
PROOF.
W'e owe this proof to Loren D. Pitt.
Note that X is mean square
continuous, since it is a sample continuous Gaussian process.
tinuous and so is every f€R(R)
0
Assume
(81).
Thus R is con-
Then, by the sample continuity
of X and the continuity of f n '
as ctoo, and hence there exists c>O
(5)
such that
now follows from the equivalence of the Gaussian processes X and X+cfn
61
(since cfn
€
o
R(R) ).
We next show that
(8 1)
is satisfied by all processes with stationary
increments having rational spectral densities.
(81 )
This implies in particular that
is satisfied by the Wiener process, by stationary processes with rational
spectral densities> and by (indefinite) integrals of stationary processes with
rational spectral densities.
The proof is based on the following result which is
of independent interest.
J..EI-1AA 5.5.
A mean squaroe aontinuou8 prooaess X =
mean and aovaroianae C(t,s)
measUr'e
{~>
-oo<t<oo} with zeroo
has (wide sense) stationaroy inaroements (with speatrat
dF) if and onz,y if theroe is a mean squaroe aontinuou8 measUr'abte (wide
sense) stationaroy prooaess Y = {Yt > -oo<t<oo} with zero mean and aovaroianae ret,s)
(with speotraz, measUr'e
dF)
suah that foro eaah t
and s
(5.5)
and
H(~X)
= H(Y).
AZ,so if XO=O a.s., then
HiZ,bert spaae of C, if and onZy for some
f(t) = get) - g(O) -
(5.6)
PROOF.
feR(C)3 the reprooduaing kemeZ,
geR(r) and atZ t
f:
g(u)du ·
The sufficiency of (5. 5) being obvious> l'1e only show its necessity.
Suppose that X has the spectral representation given by (1.2) and (1.3).
we have
Then
62
and H(L\X) = H(L\V).
(A+i)dUA =i~I(1+A2)~ elVA' Then EidUAI = EldVAI = dF(AL
and H(L\U) = H(L\V). Define the process Y = {Y ~ -co<t<ao} by Y = feitAdU .
t
t
A
Then Y is (wide sense) stationary and mean sqtlare continuous ~ and hence it has
U=
{UA~
Define the process with orthogonal increments
-ao<A<ao}
a measurable
modification~
H(L\V) = H(L\X).
2
2
by
denoted again by Y.
.Also for each fixed t
First we see that H(Y)
~le
s
and
= H(L\U)
have the following equalities
The last equality is justified by the following equality for all v ~
=I
co
-co
It
ei(U-V)Adu dF(A)
s
= It ~
ei(U-V)AdF(A) du
s Lao
=
It
E(Y Y ) du
uv
s
= E(It Yudu
s
0
Y )
v
where Fubini v s theorem has been repeatedly applied, its justification being quite
clear.
n::L Z and t
For the second claim. notice that for all
(5.7)
E(Xj,TU
= E(YtTU
- E(YOTU -
f:
E(YuTUdu •
~"e have by (5.5)
:::
63
tit
If fe:R(C)
ne:L Z
EO~ru
-f(t) = E(Xtn)
= E(Xtrv
with get)
some
then
and thus
1El,~,JA
ComrerselY1 if g€R(r)
€ R(r).
if f
TriUS
ne:L Z and (5.6) follo'VJs from (5.7)
for some
satisfies (5:5)
then
it follows from (5.7) that
X be a zero mean mean squar-e aontinuous pr-oaess with (wide
sense) stationar-y inorements having a r-ationaZ speatraZ
anae C.
Then oondition
PROOF.
Thus
d£nsity~
and with aovar-i-
is satisfied.
(8 )
1
X is as in LerrllD.a 5.5, and elF
then \'lell 1-:noT~m that
R(r) = 1d~
,
has a rational density.
integrable m-th tierivatives (with
n~l,
by a suitable choice of
have the desired property stated in
and square
2m:: degree of denominator - degree of nvmera-
tor of the polynomials of the rational spectral density).
for each fixed
It is
the set of all functions possessL."1g on every
finite interval absolutely conti.'llUOUS d.erivatives up to order m-I
It
f(t) =
o
feRce).
5.6. Let
for
get) = EcYtru
(Sl).
g€l'~?
Indeed~
f
Now it is clear that
defined by (5.6) ",-lill
for fixed
n~l
we may choose
in lf~ satisfying (8 ) and £(0)=0. Then a siq}le calculation shows that g
1
t
defined by get) = e f~e -U:fV (u)du belongs to i::~l and satisfies (S. 6). Hence~ by
f
Lemma 5.5,
fen(C).
Since
f
was chosen to satisfy
(81)
the proof is cOTIrplete.
o
Finally we reiTIarlc that Hiener's (1953) discussion of the analysis and
Sy11-
thesis of nonlinear net\'JOrks with \,-lhite noise input remains valid with only minor
1l1odifications for nonlinear networks with input a zero mea'll Gaussia.,'1. process with
stationary hlcrements.
64
6.
APPEWJI)(
We state here the basic definitions and properties of tensor products of
Hilbert spaces (see for instance Neveu (1968) ) and of Hennite polynomials which
are used throughout the paper.
6.1. Tensor Product of Hilbert Spaces
Let HI
<0 ~ 0>2.
and HZ
be two Hilbert spaces with inner products
For each hI e:Hp
for all
{r~ ~n) 8h~n): hin ) e:Hp h~n) e:HZ' N:2:l} then
If H =
and
hZE:H Z define the map h18hZ: HI xHZ-+R by
= <~,gl>1<h~2,gZ>2
(h18 hz)(gl,gZ)
<o~o>l
gl€Hl , gZe:HZ •
H is an mTler product space
under the inner product
<A B>
~
where A
and HZ
T.r.'
in HZ
and
= r~ ,~n) 8h~n),
=
I'll
r
2 <hen) gem»~ <hen) gem»~
n=l m=l I '1 1 2 '2 Z
N
B = IiI gim) ~g~m).
The tensor product HI8HZ o£ HI
is the completion of H with respect to its inner product.
{fa- a,e:A}
is
CO)::J}lete
then {fa,8g(3' a,e:A, (3e:B}
(XZ,SZ"llZ)
(COl'iS) ill HI
and
{f:f3~
(3e:B}
is complete (CONS) in BI 8HZ •
is corrq;>lete (C.DNS)
If
(Xl'SI'lll)
are two measure spaces, then LZ(XpSl'lll) 8 LZ(Xz,SZ,llz) ;;
LZ(X1xXz, SlxS Z' lllxll Z)
with corresponding elements under the isomorphism £18 £2
and fl(xl)fZ(x Z)'
Similarly the tensor product H10 ••• ~~ is defined for p Hilbert spaces
HI' ••. ~H.P.
The inner product is such that
65
and all properties carry over from the case p=2
,J'f or ~p for
~
in
If Hl =.• '=1>=H~ we write a:J>H or Hep for He ••• eH and
f8 ... ef.
an obvious manner.
For each
to the case of general p
= (~lg ..• ,~p)
e IT
p , the set of all permutations of (l, .•• ,p),
there is a unique unitary operator U~ on J1-1 such that for all f1"" gfpd-I,
U~(fle ... efp ) = f~ 13:' •• @f~ • An element f in ePH is called symmetric if
,
I
P
,..
U~f=f for all ~eITp' and the synmetric tensor product space ePH or ffP is
the set of all symmetric tensors in ePH.
the operator
(p!flI~U~ is the projection operator onto ~. Hence
= sp{
@PH
If
;PH is a closed subspace of @PH mld
(X,S,ll)
f;,- I
~
f~ el... ef~,
1
f1''' . ,fp€l-I } •
P
is a measure space and .L2(XP,SP~llP)
functions in LZ(XP ,SP ,1lP)
then f?Lz(X,S,ll);;
(Le.,
1Z(XP ,SP,llP).
= f(x~
f(xl''' .,xp )
Finally if
then {e el... ele : Yl, ••. ,yp€r} and {e
Yl
yp
YI
sets in ;PH and ;PH respectively.
the set of all symmetric
1
y , yef}
{e
, ... ,x~)
P
for all ~e~ L
is a complete set in Fl,
e... @ey:yl,
... ,yp€r}arecomplete
P
6.2 Hermite Polynomials
Let X be a Cklussian variable with zero mean and variance t.
.Applying
the Gram-Schmidt procedure to orthogonalize the sequence of random variables
1 ,A, xZ, x3 , ..• in LZ(X), we obtain the orthogonal sequence HO,t(X), Hl,t(X),
'\7
Hz,t(XL ....
~,t
is called the Hermite polynomial of degree p
with parameter t, and is a polynomial in both variables t
few Hermite polynomials are
and X.
= 0,1,2, •••
The first
,
66
•• t COr)
Z, A
.ti
= )7\.Z-t
The Hermite polynomials satisfy the following properties
67
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w.
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.;}]
It6, K. (1956).
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~
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9]
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0]
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ential
Analysis II.
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H. P. (1973).
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Adison-Wesley, Reading, Hass.
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de
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