Conditional Expectation of White
Noise Functionals
by
Yu-Chun Lin
Advisor
Yuh-Jia Lee
Institute of Statistics,
National University of Kaohsiung
Kaohsiung, Taiwan 811 R.O.C.
June 2006
Contents
1 Introduction
1
2 Preliminaries
2
3 The Schwartz Space as a nuclear space
3
4 Test and Generalized White Noise Functionals
4
5 Conditional Expectations of GWNF
6
6 Example
10
References
14
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Conditional Expectation of White Noise
Functionals
Advisor: Dr. Yuh-Jia Lee
Department of Applied Mathematics
National University of Kaohsiung
Student: Yu-Chun Lin
Institute of Statistics
National University of Kaohsiung
Abstract
In this paper it is show that the conditional expectation of a white noise functional
ϕ given the the Brownian motion B(t) is represented by
Z
E[ϕ|Bt ] =
ϕ[Θt x + (1 − Θt )y]µ(dy) ,
S∗
where Θt is the Heaviside function
(
Θt (s) ≡
I , s ≤ t,
0 , s > t.
and Θt is the Heaviside operator defined by Θt x(s) = Θt (s)x(s).
Note that the Brownian motion B(t) can be represented by
(
hx, 1[0,t] i , t ≥ 0, x ∈ S ∗ (R1 );
Bt (x) =
−hx, 1[t,0] i , t < 0, x ∈ S ∗ (R1 ).
If {ej : 1 ≤ j ≤ n} be an orthonormal set in L2 (R1 ) and Bn = σ{hx, ej i : 1 ≤ j ≤ n}
and if Pn denotes the orthogonal projection of L2 (R1 ) onto the space spanned by {ej : 1 ≤
j ≤ n}, then it is shown that conditional expectation enjoy the integral representation
Z
E[ϕ|Bn ] =
ϕ[Pn x + (1 − Pn )y]µ(dy)
S∗
Using the above integral representation we are able to investigate the regularity
properties of the conditional expectation and compute the conditional expectation easily.
Moreover, we can extend the concept of conditional expectation to generalized white
noise functionals. As applications, we give some examples.
Key words: abstract Wiener space, Brownian motion, Fréchet derivative,
Gel’fand triple, generalized functions, second quantization
iii
1
Introduction
First, a very quick review of the analysis of white noise functionals is presented. Let
(Ω, F, P) be a probability space and G is a sub-σ-field of F. In this paper, we concern
ourself about the smoothness of the conditional expectation E[X|G] for a random variable
X which is measurable with respect to the Borel field generated by the Brownian motion.
Generally speaking, to study such a problem is almost impossible without further knowing
the structure of Ω. For such a purpose, we choose as our underlying probability space
(S ∗ , B, µ)(B is the Borel field on S ∗ ), where S denoted the Schwartz space over R1 and S ∗
is the dual space of S, also known as the space of tempered distributions. S is a nuclear
space and S ⊂ L2 (R1 ) ⊂ S ∗ forms a Gel’fand triple. It is well known that S ∗ carries a
Gaussian measure µ which is characterized as the unique probability measure such that,
for all ξ ∈ S,
Z
1
ei(X,ξ) µ(dx) = e− 2 |ξ|
2
S∗
where (·, ·) denotes the S ∗ − S pairing and | · | denotes the L2 (R1 )-norm.
The elements of S ∗ are regarded as the sample paths of the white noise process and
the space (S ∗ , µ) is referred to as a White Noise space. Then the Brownian motion B(t)
can be represented by
(
Bt (x) =
hx, 1[0,t] i , t ≥ 0, x ∈ S ∗ (R1 );
−hx, 1[t,0] i , t < 0, x ∈ S ∗ (R1 ).
Using this representation, one can further represent the conditional expectation E[ϕ|Bt ]
for a analytic functional ϕ ∈ (S) given by the following integral
Z
ϕ[Θt x + (1 − Θt )y]µ(dy)
E[ϕ|Bt ] =
(1.1)
S∗
We shall also define and study the conditional expectation for generalized white
noise functional. There the space of test functions is taken to be the space (S). (We will
discuss this in detail in a later chapter.) Members of the dual space (S)∗ of (S) will be
called the Generalized White Noise Functionals (GWNF, for abbrev.). This also provides
a suitable framework for white noise analysis. For example, the white noise functional
hhḂ(t1 ) · · · Ḃ(tn ), ϕii can be defined by the linear functional Dn µϕ(0)δt1 · · · δtn where ϕ
is an arbitrary test function, µ is the Canonical Gaussian measure on the space S ∗ of
R
tempered distribution, where µϕ(x) = S ∗ ϕ(x+y)µ(dy), D denotes the Fréchet derivative
and δt the Dirac measure concentrated at t. From the integral representation (1.1) of
conditional expectation one can show that E[ϕ|Bt ] ∈ (S) for any ϕ ∈ (S) and E[·|Bt ]
1
defines a continuous operator on (S). Then the conditional expectation of generalized
white noise functionals are defined by the relation:
hhE[Φ|Bn ], ϕii = hhΦ, E[ϕ|Bn ]ii
(1.2)
for Φ ∈ (S)∗ and ϕ ∈ (S).
The contents of this paper are arranged as follows: In §2, we describe briefly the
background concerning the concepts of nuclear space and Gel’fand triple. In §3 and §4,
we briefly introduced respectively a reconstruction of the Schwartz space and the theory
of white noise functionals. In §5, we first prove the integral representation of conditional
expectation, then employing the representation we are able to study the regulity properties of conditional expectation and extend the concept of conditional expectation to
white noise functionals in §6.
2
Preliminaries
The underlying infinite dimensional space is described by a Gel’fand triples which
will be introduced briefly as following. In this section we describe briefly the background
concerning the concepts of nuclear space and Gel’fand triple. For the details, see Gross
[1] or Kuo [7][8] (for abstract Wiener spaces), Gel’fand and Vilenkin [24] (for countablyHilbert spaces and nuclear spaces).
Let V be a countably-Hilbert space associated with an increasing sequence {| · |n } of
norms. Let Vn be the completion of V with respect to the norm | · |n . The countablyHilbert space V is called a N uclear Space if for any n, there exists m ≥ n such that the
inclusion map from Vm into Vn is a Hilbert-Schmidt operator, i.e., there is an orthonormal
∞
X
|vk |2n < ∞.
basis {vk } for Vm such that
k=1
Note that a trace class operator is also a Hilbert-Schmidt operator and that the
product of two Hilbert-Schmidt operators is a trace class operator. Therefore V is a
nuclear space if and only if for any n, there exists m ≥ n such that the inclusion map
from Vm into Vn is a trace class operator.
Nuclear spaces have many properties similar to those of the finite dimensional space
r
R . For instance a subset of a nuclear space is compact if and only if it is closed and
bounded. This implies that an infinite dimensional Banach space is not nuclear.
2
Fact 2.1 (Minlos theorem) Let V be a real nuclear space. A complex-valued function
ϕ on V is the characteristic function of a unique probability measure µ on V 0 , i.e.,
Z
ϕ(v) =
ei(x,v) µ(dx),
µ ∈ V,
V0
if and only if (1) ϕ(0) = 1, (2) ϕ is continuous, and (3) ϕ is positive definite,
i.e., for any z1 , z2 , · · · , zn ∈ C and µ1 , µ2 , · · · , µn ∈ V ,
n
X
zj z̄k ϕ(µj − µk ) ≥ 0.
j,k=1
Fact 2.2 (Gel0 fand triples)
Clearly, V is imbedded in E and that V is dense in E relative to the norm of E. By
using the Riesz representation theorem to identify E with its dual space E 0 , we get the
triple
V ⊂ E ⊂ V 0.
such a triple is called a Gel’fand triple. Note that E is dense in V 0 with the weak topology
of V 0 .
3
The Schwartz Space as a nuclear space
In this section we will reconstruct the Schwartz space S(R1 ) from the real Hilbert
space L2 (R1 ) and denote the self-adjoint operator A = −d2 /dx2 + x2 + 1.
The operator A is densely defined on L2 (R1 ) and A has eigenvalues {2n + 2 : n =
0, 1, 2, · · ·} with corresponding eigenfunctions {en } given by the Hermite functions. Define
|f |p = |Ap f |, where p ∈ R, f ∈ D(Ap ) and | · | denotes the norm of L2 (R1 ). If p ≥ 0,
| · |p ≥ | · |; if p < 0, | · |p < | · | [18]. For p ≥ 0, let Sp = D(Ap ). Then (Sp , | · |p ) is a Hilbert
space with dual given by (S−p , | · |−p ) which is defined by the completion of L2 (R1 ) with
respect to | · |−p . It is easy to see that S = ∩p≥0 Sp , S ∗ = ∪p≥0 S−p , and S and S ∗ are
topologized respectively by the projective limit of {Sp }, and the inductive limit of {S−p }.
If p > 21 , then (L2 , S−p ) forms an abstract Wiener space. By the Minlos theorem, it is
well-known that S carries a standard Gaussian measure µ with characteristic function
Z
1
2
C(ξ) =
ei(x,ξ) µ(dx) = e− 2 |ξ|
S∗
for ξ ∈ S, and µ(S−p ) = 1 for p > 12 , where (·, ·) denotes the dual pair of (S ∗ , S) and
(S−p , Sp ).
3
Definition 3.1 The probability space (S ∗ (R1 ), µ) is called a White Noise Space which
will serves as the underlying space in our investigation. The measure µ is called the
standard Gaussian measure on S ∗ (R1 ).
The reason for calling (S ∗ (R1 ), µ) a white noise space is as follows. Note that for
each ξ ∈ S(R1 ), the random variable h·, ξi is defined everywhere on S ∗ (R1 ). It is normally
distributed with mean 0 and variance |ξ|2 . Now, suppose f ∈ L2 (R1 ) (real-valued). Take
a sequence {ξn } in S(R1 ) such that ξn → f in L2 (R1 ). Then the sequence {h·, ξn i} of
random variables is Cauchy in L2 (S ∗ (R1 ), µ). Define h·, f i = lim h·, ξn i in L2 (S ∗ (R1 ), µ).
n→∞
The limit is independent of the sequence. Moreover, the random variable h·, f i is normally distributed with mean 0 and variance |f |2 .
Definition 3.2 Define the stochastic process
(
hx, 1[0,t] i , t ≥ 0, x ∈ S ∗ (R1 );
Bt (x) =
−hx, 1[t,0] i , t < 0, x ∈ S ∗ (R1 ).
(3.1)
It is easy to check Bt is a Brownian motion. Moreover, by taking the time derivative
informally, we get Ḃt = x(t). Thus the elements of S ∗ (R1 ) can be regarded as the sample
paths of white noise.
4
Test and Generalized White Noise Functionals
We will use the Wiener-Itô theorem to construct the test function.
Wiener Itô theorem
Let In be the multiple Wiener integral of order n with respect to the Brownian
R∞
R∞
c2 (Rn ). L
c2 (Rn )
motion given by In (ϕ) = −∞ · · · −∞ ϕ(t1 , · · · , tn )dB(t1 ) · · · dB(tn ), ϕ ∈ L
denotes the space of symmetric complex-valued L2 -functions on Rn . Any function f ∈
∞
X
2
2
∗
n
2
c
(L ) = L (S , µ), existing fn ∈ L (R ), can be written as f =
⊕In (fn ) and
kf k2L2 =
∞
X
n=0
n! |fn |2L2 .
n=0
We list below some notations used frequently in this thesis.
(1) D denotes the Fréchet derivative in the direction of L2 .
(2) µf (x) =
R
S−p
f (x + y)µ(dy) .
(3) Ln(2) [S−p ] : the space of all n-linear Hilbert-Schmidt operators on S−p .
4
(4) T ∈ Ln(2) [H], λn1 T (x) =
R
S∗
T (x + iy)n µ(dy).
For the existence and more properties of λn1 T (x), we refer the reader to [22].
Theorem 4.1 (A Reformulation of Wiener-Itô Theorem)
Let f ∈ L2 (µ) and define Qn f =
1 n
λ
n! 1
[Dn µf (0)]. Then we have
(a) Qn is an orthogonal projection from L2 (µ) onto the homogeneous chaos Hn where is
defined as the collection {λn1 T : T ∈ SLn(2) [H]} and hh·, ·ii the L2 (µ) inner product.
Z
1 n n
f (x)
λ [D µϕ(0)](x) µ(dx)
hhQn f, ϕii = hhf, Qn ϕii =
n! 1
S∗
Z 1 n n
1
n
n
λ [D µf (0)](x) ϕ(0)µ(dx)
= hhD µf (0), D µϕ(0)iiHS =
n!
n! 1
S∗
1
= hh λn1 [Dn µf (0)], ϕii
n!
where hhS, T iiHS =
∞
X
(Sei1 · · · ein )(T ei1 · · · ein ), {ek } is an orthonormal basis of H.
i1 ,···,in =1
(b) f =
∞
X
n=0
(c)
kf k2L2 (µ)
⊕
1 n n
λ [D µf (0)].
n! 1
∞
X
1
=
kDn µf (0)k2HS .
n!
n=0
For p ≥ 0 and for f ∈ (L2 ), define
kf k2,p
(∞
) 12
X 1
=
kDn µf (0)k2Ln [S−p ]
(2)
n!
n=0
Let Γ(Ap ) denote the second quantization of Ap . For p ≥ 0, we set (Sp ) = D(Γ(Ap )).
(Sp ) denote the collection of function f in (L2 ) for which kf k2,p is finite and let (S) =
∩p∈N (Sp ). (S) is provided with the projective limit topology. The members of the dual
space (S)∗ are called generalized White Noise functionals or Hida distributions. Moreover,
(S) ⊂ (L2 ) ⊂ (S)∗ . The basic machinery in white noise analysis is called S-transform
which is used to define and study the GWNF. Define S-transform SF of F as follows:
(
µF (ξ)
, ξ ∈ L2 , F ∈ (L2 );
SF (ξ) =
1
2
e− 2 |ξ| hhF, e(·,ξ) ii , ξ ∈ S, F ∈ (Sp )∗ .
where hh·, ·ii denotes the (S)∗ − (S) pairing and (·, ·) the S ∗ − S pairing.
If hh·, ·iip denotes the (Sp )∗ − (Sp ) pairing, it is easy to see that, for F ∈ (Sp )∗ and
ϕ ∈ (S), we have hhF, ϕii = hhF, ϕiip . Thus, for notational convenience, we shall use hh·, ·ii
5
to denote the (Sp )∗ − (Sp ) as well. Similarly, we shall also use (·, ·) to denote the S−p − Sp
pairing. From the above, one see easily that SF (ξ) is defined also for ξ ∈ Sp if F ∈ (Sp )∗ .
More precisely, the S-transform transforms a GWNF F in a unique way to a U functional UF (ξ) defined on the Schwartz space S. Then the GWNF F is defined by
S −1 UF . In Leeś previous paper, it was shown that the GWNF could be also defined and
studied in terms of their linear functional forms without using inverse S-transform.
The smoothness of test functions play a important role in Leeś investigation. In[17],
he showed that (S) has a analytic version A∞ . It is also shown that, A∞ is an algebra as
well as a topological linear space which is topologized as the Banach spaces Ap of analytic
1
2
functions on CS−p (p > 0) with norm given by kf kAp = supz∈CS−p {|f (z)|e− 2 kzk−p }, where
CS−p denotes the complexification of S−p . Moreover, it is shown that the three family of
norms {k · k2,p }, {kS(·)kAp }, and {k · kAp } are equivalent on A∞ . Therefore, {Ap , k · kAp }
is a Banach space. For all q > p ≥ 0, we get Ap ⊃ Aq . Let A∞ = ∩p≥0 Ap and A∞ is
endowed with projective limit topology, then A∞ is a complete topological linear space.
5
Conditional Expectations of GWNF
From the previous paper[24], recall that let (H, B) be an abstract Wiener space
and pt be an Wiener measure with variance parameter t > 0. Let Pt (x) = x(s ∧ t) be
a continuous projection of B and Qt = I − Pt . Then Q is also a continuous projection
of B. Let kerT be the kernel of the operator T . Then kerP = QB and P B = kerQ are
closed subspaces of B, and B = P B ⊕ QB.
Assume further that the restriction of P on H is an orthogonal projection. Then
B = P B × QB and we have
pt = p0t × p00t
(5.1)
where pt , p0t and p00t are respectively the abstract Wiener measure on B, P B and QB with
variance parameter t.
Proposition 5.1 f : B → R is measurable function, we have
Z
Z Z
f (z)pt (dz) =
f (P x + Qy)pt (dx)pt (dy)
B
B
(5.2)
B
provided that both integrals exist. Furthermore, if f ∈ L1 (B, pt ), then the function
(x, y) → f (P x + Qy) ∈ L1 (B × B, pt × pt ).
6
Proposition 5.2 Let h ∈ H and α ∈ C, then we have
Z
1 2
2
eα(x,h) ϕ(x)p1 (dx) = e 2 α |h| p1 ϕ(αh)
(5.3)
B
Theorem 5.3 Let {ej : j = 1, 2, · · · n} be an orthonormal set in L2 (R1 ) and let Bn the
n
X
σ-field generated by {ẽj : 1 ≤ j ≤ n}. Let Pn (x) =
hx, ej iej . Then for ϕ ∈ (L2 ),
j=1
we have
Z
E[ϕ|Bn ] =
ϕ(Pn x + Qn y)µ(dy)
S∗
Proof. In fact, for λ1 , λ2 , · · · , λn ∈ R, we have
(
)
Z
n
X
E [ϕ| Bn ] exp −i
λj eej µ(dx)
S∗
j=1
" "
(
) ##
= E E ϕ(x) exp −i
λj eej Bn
j=1
)#
"
(
n
X
= E ϕ(x) exp −i
λj eej
n
X
j=1
(
Z
ϕ(x) exp −i
=
S∗
n
X
)
λj hx, ej i µ(dx)
[ by (5.3)]
j=1
(
n
1X 2
= exp −
λ
2 j=1 j
)Z
ϕ x+i
S∗
n
X
!
µ(dx)
λj ej
j=1
On other hand,
(
)
Z Z
n
X
ϕ(Pn x + Qn y)µ(dy) exp −i
λj eej µ(dx)
S∗
S∗
Z
(
Z
ϕ(Pn x + Qn y) exp −i
=
S∗
S∗
(
[ by (5.3)]
j=1
n
X
)
λj eej
µ(dy)µ(dx)
[ by (5.2)]
j=1
n
X
)Z
n
X
1
2
= exp −
ϕ Pn x + Qn y + i
λ
λj ej
2 j=1 j
S∗ S∗
j=1
)Z
!
(
n
n
X
1X 2
ϕ x+i
= exp −
λ
λj ej µ(dx)
2 j=1 j
S∗
j=1
Z
!
µ(dy)µ(dx)
We have proved that
Z
E[ϕ|Bn ] =
ϕ(Pn x + Qn y)µ(dy)
S∗
7
f or a.a x ∈ S ∗
Lemma 5.4 Given ϕ ∈ Lα , 1 ≤ α < ∞, and ε > 0, there is a bounded measurable
function ϕM and kϕ − ϕM kLα < ε.
(a) ϕM ∈ L∞ ⊂ Lα , k ϕM − ϕ kLα → 0.
(b) {E[ϕM |Bn ]} is a Cauchy sequence.
Proof.
(a) Let
ϕN (x) =



 N
, N ≤ ϕ(x),
ϕ(x) , −N ≤ ϕ(x) ≤ N,


 −N , ϕ(x) ≤ −N.
Then |ϕN | ≤ N , and hϕN i converges to ϕ everywhere, and so |ϕ − ϕN |α → 0. Since
R
|ϕ − ϕN |α ≤ 2|ϕ|α , |ϕ|α is integrable, we have k ϕ − ϕN kαLα = |ϕ − ϕN |α → 0 as
N → ∞. Thus k ϕ − ϕN kLα → 0, and there is an M with k ϕ − ϕM kLα < ε.
(b) Since E[ϕ|Bn ] =
R
S∗
ϕ(Pn x + Qn y)µ(dy). For α ≥ 1, employing Jensen inequality
, we have k ϕM − ϕM 0 kLα ≥ k E[ϕM |Bn ] − E[ϕM 0 |Bn ] kLα , there exists M 0 ∈ N0 ,
then E[ϕM |Bn ] → E[ϕ|Bn ] exists, as M → ∞. Consequently, E[ϕM |Bn ] is a Cauchy
sequence. Hence E[ϕM |Bn ] converges to E[ϕ|Bn ].
R
From 5.4, S ∗ ϕ(Pn x + Qn y)µ(dy) makes sense and may be defined by the following.
Definition 5.5
Z
Z
α
∗
ϕ(Pn x + Qn y)µ(dy) = L (S , µ) − lim
M →∞
S∗
ϕM (Pn x + Qn y)µ(dy).
(5.4)
S∗
as 1 ≤ α < ∞.
Theorem 5.6 Define the map Λn on (S) by Λn (ϕ) =
R
S∗
ϕ(Pn x + Qn y)µ(dy) a specific
version of E[ϕ|Bn ]. Then Λn (ϕ) ∈ (S) and Λn is a bounded operator on (S). Moreover,
hhΛn (ϕ), ψii = hhϕ, Λn (ψ)ii, for all ϕ, ψ ∈ (S).
Proof. For any ϕ ∈ (S) = A∞ , ϕ ∈ Ap for all p ∈ N,
Z
1
2
|Λn (ϕ)(x)| ≤
|ϕ(Pn x + Qn y)|µ(dy) ≤ C · kϕkAp · e 2 |x|−p
(5.5)
S∗
where C =
R
1
S∗
2
e 2 |y|−p µ(dy) < ∞ by Férnique’s theorem. For any x, z ∈ S ∗ and any closed
cure γ in the complex plane,
Z
Z Z
Λn (ϕ)(x + λz)dλ =
γ
γ
ϕ[Pn (x + λz) + Qn y]µ(dy)dλ = 0
S∗
8
By Morera’s theorem, Λn (ϕ)(x + λz) is entire function of complex variable λ.
Therefore Λn (ϕ) ∈ A∞ . Now it follows from (5.5) that
kΛn (ϕ)kAp ≤ C · kϕkAp
This shows that Λn is continuous on A∞ = (S). Moreover, since (S) ⊂ (L2 ) and since
conditional expectation Λn is an orthogonal projection on (L2 ), we have
hhΛn (ϕ), ψii = hhϕ, Λn (ψ)ii , for ϕ, ψ ∈ (S).
Theorem 5.6 leads us to the following definition.
Definition 5.7 The conditional expectation of generalized white noise functionals are
defined by the relation: hhE[F |Bn ], ϕii = hhF, E[ϕ|Bn ]ii, for F ∈ (S)∗ and ϕ ∈ (S).
Definition 5.8 (Contraction)
Let X = (X, d) be a metric space. A mapping T : X → X is called a contraction
on X if there is a positive real number β ≤ 1 such that for all x, y ∈ X
d(T x, T y) ≤ βd(x, y)
Geometrically this means that any points x and y have images that are closer together
than those points x and y; more precisely, the ratio d(T x, T y)/d(x, y) does not exceed a
constant β which is strictly less than 1.
2
∗
Theorem 5.9 It has been shown in [19]. If ϕ ∈ L (S , µ), define Pn (x) =
n
X
hx, ej iej ,
j=1
and if K is a contraction from S−p to L2 (p ≥ 0), then we have
Z
√
2
∗
Γ(K)ϕ(x) = L (S , µ) − lim
ϕ[K ∗ Pn x + ( I − K ∗ K)Pn y]µ(dy).
n→∞
(5.6)
S∗
where K ∗ is the adjoint of K.
In particular, K is a contraction on L2 , K is a self-adjoint operator.
√
R
Then Γ(K)ϕ(x) = S ∗ ϕ(Kx + I − K 2 y)µ(dy).
R
√
Thus, S ∗ ϕ(K ∗ x + I − K ∗ Ky)µ(dy) make sense and it is understood as limit (5.6).
Remark 5.10 Γ(Θt ) is a bounded linear operator from the space of regular generalized
functions (S)∗ into itself. Γ(Θt )ϕ coincides with the conditional expectation for elements
ϕ from (S)∗ with respect to Bt . (For detail, we refer to [20].)
Γ(Θt )ϕ = E[ϕ|Bt ]
9
By using the same arguments as given in the proof of Theorem 5.3, or by Remark 5.11
we have, by [20], one can show that
Z
p
E[ϕ|Bt ] =
ϕ(Θt x + I − Θ2t y)µ(dy)
S∗
Let Rt =
p
I − Θ2t , and Rt2 = I − Θ2t = I − Θt = Rt .
Then
Z
ϕ[Θt x + (I − Θt )y]µ(dy)
E[ϕ|Bt ] =
S∗
6
Example
We conclude this paper by given some example.
Example 6.1 Let h ∈ L2 (R1 ) with |h| =
6 0. Then we have
(a) E[B(t)|e
h] = hht , hi, where e
h(x) = hx, hi, B(t) = het , and
(
1[0,t] , t ≥ 0,
ht =
−1[t,0] , t < 0.
Proof.
Z
hP x + Qy, ht iµdy
E[B(t)|e
h] =
(P (x) = hx, h/|h|ih/|h|)
∗
ZS
(hx, P ht i + hx, Qht i)µdy
Z
1 e t
h
h(s)ds
=
|h|2 −∞
=
S∗
In particular, as h = hs , we have
(b) For s ≤ t, E[B(t)|B(s)] = B(s) which coincides with the result by using the classical
probability theory.
(c) If h is a non-zero continuous function such that h ∈ L2 (R1 ) , then
1
E[Ḃ(t)|e
h] = 2 h(t)e
h
|h|
10
Proof. For ϕ ∈ (S) and let P be as define in (a), then we have
Z
hhE[Ḃ(t)|e
h], ϕii = hhḂ(t), E[ϕ|e
h]ii = Dµ
ϕ(P x + Qy)µ(dy) (0)δt
S∗
hDµϕ(P ξ), yi = hDµϕ(P ξ), P yi
Replacing y by δt , and let P ξ = 0
hDµϕ(0), P δt i
Z
=
ϕ(x)hx, P δt iµ(dx)
∗
ZS
ϕ(x)hδt , hihx, hiµ(dx)
=
S∗
= hhhδt , hie
h, ϕii
1
Therefore E[Ḃ(t)|e
h] = hδt , hi · e
h = 2 h(t)e
h.
|h|
Remark 6.2 We remark that
d
E[B(t)|e
h]
dt
= E[Ḃ(t)|e
h].
The next example concerning conditional expectation require Kac formula. For the sake
of clarity we first give a proof of Kac formula by white noise approach.
Example 6.3 (Derivation of Kac formula via white noise calculus)
For α < π 2 /8, we have
Z
E exp α
1
q
√
B(t) dt
= csch 2α.
2
0
(a) Note that for x, y ∈ S ∗ ,
Z 1
Z
hx, 1[0,t] ihy, 1[0,t] idt =
0
1
0
Z
1
(1 − t ∨ s)x(s)y(t)dsdt.
0
Proof.
Z 1
hx, 1[0,t] ihy, 1[0,t] idt
Z t
Z t Z s
1 Z 1 Z t Z s
=
x(s)ds
y(u)du ds −
y(u)du ds x(t)dt
0
0
0
0
0
0
0
Z 1
Z 1 Z s
Z 1 Z t Z s
=
x(s)ds
y(u)du ds −
y(u)du ds x(t)dt
0
0
0
0
0
0
Z 1 Z 1 Z s
=
y(u)du ds x(t)dt
0
t
0
Z 1 Z 1
Z 1
=
y(u)du −
(s ∨ t)y(s)ds x(t)dt
0
0
0
Z 1Z 1
=
(1 − s ∨ t)y(s)x(t)dsdt
0
0
0
= hAy, xi = hy, Axi.
11
where
R1
0
(t ∨ s)y(s)ds =
R1
0
y(s)ds −
R1Rs
0
t
y(u)duds; and hx, yi =
R∞
−∞
x(t)y(t)dt.
A is clearly self-adjoint positive operator on L2 . To find the eigenvalues of A, we
go into details on next step.
(b) A is give by
Z
1
(1 − s ∨ t)x(s)ds
Ax(t) =
(6.1)
0
whose eigenvalues are given by
(
)
2
2
: n = 0, 1, 2, . . .
(2n + 1)π
Proof. It follows from (a) that it is easy to see that A is given by (6.1). To find
the eigenvalues of A, we consider the equation
Ax(t) = λx(t)
(6.2)
we write
Z
t
Z
(1 − t)x(s)ds +
Ax(t) =
0
1
(1 − s)x(s)ds .
t
It is immediately observed that x should satisfies x(1) = 0. Differentiating (6.2) we
obtain
Z
0
λx (t) = −
t
x(s)ds ,
0
and differentiating (6.2) twice, we obtain
λx00 (t) = −x(t) .
This implies that x0 (0) = 0 (λ = 0 is not an eigenvalue).
Thus the eigenvalue problem (6.2) leads to the following boundary value problem
(
λx00 (t) + x(t) = 0,
(6.3)
x(1) = x0 (0) = 0. (λ > 0)
It is easy to find the solutions, λ is an eigenvalue iff cos √1λ = 0. Therefore λ are
∞
h
i2
X
2
given by λn = (2n+1)π , n = 0, 1, 2, · · ·. Since
λn < ∞, A is a trace class
n=0
R1
operator and hAx, xi = 0 x2 (t)dt for almost all x.
We obtain the result on the next page,
12
1
Z
E exp α
Z
B(t) dt
=
1
exp (αhAx, xi) µ(dx) = [det (I − 2αA)]− 2
2
S∗
0
"
=
∞
Y
#− 12
(1 − 2αλn )
n=0


∞
Y
1 −
=
n=0
!2 − 12

8α


(2n + 1)π
√
q
√
=
csch 2α
Example 6.4 Let h ∈ L2 (R1 ) such that supp(h) ⊂ [0, 1] and |h| = 1.
Let ϕ(x) = exp [αhAx, xi], P (x) = hx, hih, and Q = I − P . Then for α < π 2 /8, we have
Z 1
2
h
B(t) dt e
E exp α
0
i
h
= E eαhAx,xi e
Z
=
ϕ(P x + Qy)µ(dy)
∗
S
Z
exp {αhA(P x + Qy), (P x + Qy)i} µ(dy)
=
∗
S
Z
=
exp {α[hAP x, P xi + 2hAP x, Qyi + hAQy, Qyi]} µ(dy)
S∗
Z
αhAP x,P xi
=e
exp{α[2hAP x, Qyi + hAQy, Qyi]}µ(dy) (According to (5.1)(5.2), let y 0 = Qy.)
∗
ZS
= eαhAP x,P xi
exp{α[2hAP x, y 0 i + hAy 0 , y 0 i]}µ(dy 0 )
∗
QS
!
Z
∞
X
= eαhAP x,P xi
exp{α[2hAP x, yi + hAy, yi]}µ(dy)
Employing hAx, yi =
λi hx, ei ihy, ei i
S∗
= eαhAP x,P xi
i=1
(
Z
exp α
S∗
∞ Z
Y
∞
X
2λi hx, ei ihy, ei i + α
i=1
∞
X
)
λi hy, ei i2
µ(dy)
i=1
∞
1
1 2
√ exp − u
exp 2αλi hx, ei iu + αλi u
du
=e
2
2π
−∞
i=1
∞ Z ∞
Y
1
1
2
αhAP x,P xi
√ exp
αλi −
u + 2αλi hx, ei iu du
=e
2
2π
i=1 −∞
(
)2
Z ∞
21
−1
∞
2 2
Y
2
−α
λ
1
1
1
i
√ exp
= eαhAP x,P xi
exp
hx, ei i2
αλi −
u + αλi −
hx, ei iαλi du
αλ
−
1/2
2
2
2π
i
−∞
i=1
(∞ ) (Y
−1/2 )
∞ 2 2
2 2
X
−α
λ
α
λ
1
i
i
= eαhAP x,P xi exp
hx, ei i2 +
hx, ei i2
αλi −
αλ
−
1/2
αλ
−
1/2
2
i
i
i=1
i=1
−1/2
1
αhAP x,P xi
=e
det αA − I
2
q
√
2
= eαhx,hi hAh,hi csch 2α
αhAP x,P xi
2
13
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