1. Stochastic integral representations of conditional laws
Our basic probability space is the one-dimensional canonical Wiener space
; F; P, equipped with the canonical Wiener process W = W
t t
∈ [0; 1]
. More precisely,
= C[0; 1]; R is the set of continuous functions on [0; 1] starting at 0, F the -algebra
of Borel sets with respect to uniform convergence on [0; 1], P Wiener measure and W the coordinate process. The natural ltration F
t t
∈ [0; 1]
of W is assumed to be completed by the sets of P-measure 0.
Guided by our prototypical example L = sup
06t61
W
t
, in this section we will give integral representations of the conditional densities of random variables L with respect
to the -algebras F
t
; t ∈ [0; 1]; of the small ltration. For the more technical basic facts
of measure-valued Malliavin calculus we refer the reader to the appendix. Let L be an F
1
-measurable random variable, and P
t
:; d x a version of the regular conditional law of L given F
t
; t ∈ [0; 1]: We know that the process P
t
:; d x; t ∈ [0; 1]; is a measure
valued martingale: for any f ∈ C
b
R; the process hP
t
:; d x; f i; t ∈ [0; 1]; is a real
valued continuous martingale which, provided L is smooth enough in the sense of Malliavin’s calculus, can be represented by the formula
hP
t
:; d x; f i = hP
:; d x; f i +
Z
t
ED
s
hP
s
+
:; d x; f i|F
s
dW
s
see Imkeller, 1996 for the setting, where all measures are absolutely continuous with respect to a joint reference measure. As follows from the martingale representation
theorem in the Wiener ltration, in order to be able to write the stochastic integral in this formula, one of course does not need Malliavin dierentiability of P
t
:; d x on the whole interval [0; t]; but just the existence of a well-behaved trace-type object
D
t
P
t
+
:; d x = lim
s ↓t
D
t
P
s
:; d x in the sense of weak ∗ convergence in L
2
× [0; 1]
2
. Not to restrict generality too much from the start, we shall work with smooth approx-
imations of P
t
:; d x and take limits only for the trace-type objects. Let L
∈ D
1; 2
; and N an additional N0; 1-variable on our probability space which is independent of F
1
: For ¿ 0; let L
= L + √
N; and P
t
:; d x a version of the regular conditional law of L given F
t
; 06t61: In this section we shall work under the hypothesis
H |D
t
L |6M;
06t61 for some random variable M the maximal function of which is p-integrable for any
p¿1, i.e. M
∗
= sup
06t61
|EM |F
t
| ∈ L
p
. Denote by p the probability density of
√ N . Then for f
∈ C
b
R we have hP
t
:; d x; f i = EfL
|F
t
= E Z
R
p y
− Lfy dy|F
t
= Z
R
Ep y
− L|F
t
fy dy =
hEp y
− L|F
t
dy; f i:
Moreover, for 06r6t; y ∈ R;
D
r
Ep y
− L|F
t
= ED
r
p y
− L|F
t
= E D
r
L L
− y p
y − L|F
t
: H allows to apply the Clark–Ocone formula, and we obtain for f
∈ C
b
R; t ∈ [0; 1]
hP
t
:; d x; f i = hP
:; d x; f i +
Z
t
E D
s
L L
− y p
y − L |F
s
dy; f dW
s
: Now dene
h
s
:; x = E D
s
L L
− x p
x − L |F
s
; k
s
:; d x = h
s
:; x d x; ¿ 0; s
∈ [0; 1]; x ∈ R. Then, due to the boundedness of x 7→ xp x and H, we
obtain a constant c such that
|h
s
:; x |6c
E |D
s
L kF
s
6c EM
|F
s
6c M
∗
; ¿ 0; s
∈ [0; 1]; x ∈ R. Therefore for f ∈ C
b
R; p¿1 E
Z
1
hk
s
:; d x; f i
2
ds
p=2
6
E
Z
1
|k
s
:; d x |
2
ds
p=2
kfk
p
6 c
p
EM
∗ p
kfk
p
: Hence we have
sup
f ∈ C
b
R; kfk61
E
Z
1
hk
s
:; d x; f i
2
ds
p=2
¡ ∞:
1 Since moreover for any ¿ 0; t
∈ [0; 1]; p¿1 by an easier argument sup
f ∈ C
b
R; kfk61
E hP
t
:; d x; f i
p
¡ ∞;
2 Proposition A.1 allows us to write
P
t
:; d x = P :; d x +
Z
t
k
s
:; d x dW
s
3 for ¿ 0; t
∈ [0; 1]: We aim at passing to the limit → 0 in 3, thereby keeping track of the convergence of the measure-valued processes k
t
:; d x; t ∈ [0; 1]: To gain a
better insight into which aspects are essential, let us rst treat our prototypical example. Our treatment shares some aspects with Jeulin’s 1980, but in contrast is based on
Malliavin’s calculus.
Example 1. Let L = sup
06t61
W
t
; N be a N0; 1-variable independent of F
1
: For t
∈ [0; 1]; 06h61−t denote by S
t
= sup
06s6t
W
s
; and
h
= sup
06k6h
W
k+t
−W
t
: Then we have
L = S
t
∨ W
t
+
1 −t
; 06t61;
with
1 −t
independent of W
t
; S
t
: Denote by f
1 −t
the density of the law of
1 −t
. We have
f
1 −t
z = r
2 1
√ 1
− t exp
− 1
21 − t
z
2
1
[0; ∞[
z; z
∈ R. It is well known see Nualart and Vives, 1988 that S
t
∈ D
1; p
for all 06t61, and, if
t
denotes the P-a.s. uniquely dened random time at which W takes its maximum on the interval [0; t]; we have
D
s
S
t
= 1
[0;
t
]
s; s
∈ [0; 1]: 4
In particular, if we omit the subscript for t = 1; we have D
s
L = 1
[0; ]
s; s
∈ [0; 1]: 5
Hence for t ∈ [0; 1]; ¿ 0; y ∈ R we obtain
Ep y
− L|F
t
= Ep y
− S
t
1
{S
t
¿W
t
+
1−t
}
|F
t
+ Ep y
− W
t
+
1 −t
1
{S
t
¡W
t
+
1−t
}
|F
t
= p y
− S
t
Z
S
t
−W
t
f
1 −t
y dy +
Z
y −S
t
−∞
p v f
1 −t
y − W
t
− v dv: Hence for r
∈ [0; t] D
r
Ep y
− L|F
t
= S
t
− y D
r
S
t
p y
− S
t
Z
S
t
−W
t
f
1 −t
y dy + p
y − S
t
f
1 −t
S
t
− W
t
D
r
S
t
− W
t
+p y
− S
t
f
1 −t
S
t
− W
t
−D
r
S
t
− Z
y −S
t
−∞
p v
y − W
t
− v 1
− t f
1 −t
y − W
t
− v dv: This in turn implies that with the above notation for t
∈ [0; 1]; x ∈ R h
t
:; x = D
t
Ep x
− L|F
t
+
= −p
x − S
t
f
1 −t
S
t
− W
t
− Z
x −S
t
−∞
p v
x − W
t
− v 1
− t f
1 −t
x − W
t
− v dv: 6
Now what happens as → 0? Let f ∈ C
b
R: Then for any t ∈ [0; 1] pointwise
h[p x
− S
t
d x −
S
t
d x]; f i =
Z
R
p x
− S
t
[fx − fS
t
] d x → 0;
7 as well as
Z
R
Z
x −S
t
−∞
p v
x − W
t
− v 1
− t f
1 −t
x − W
t
− v −1
[S
t
; ∞[
x x
− W
t
1 − t
f
1 −t
x − W
t
] dvfx d x → 0
8
as → 0. Since sup
y ∈ R
|yf
1 −t
y | ¡ ∞, this convergence is bounded by constants
depending only on t and f; and the constants are bounded on intervals [0; t] for t ¡ 1: Hence dominated convergence shows
E Z
t
h[k
s
:; d x − k
s
:; d x]; f i
2
ds → 0
9 as
→ 0; for 06t ¡ 1; where k
s
:; d x = −
S
t
d x f
1 −t
S
t
− W
t
− 1
[S
t
; ∞[
x x
− W
t
1 − t
f
1 −t
x − W
t
d x: 10
The following convergence is obvious, due to continuity. So for all 06t61; f ∈ C
b
R, we have
hP
t
:; d x; f i = EfL
|F
t
→ EfL|F
t
= hP
t
:; d x; f i:
Hence 10 yields the equation, valid for any f ∈ C
b
R; t ∈ [0; 1]
hP
t
:; d x; f i − hP
:; d x; f i =
Z
t
hk
s
:; d x; f i dW
s
: 11
The M -valued for the notation see Appendix process k:; d x even satises E
Z
t
|k
s
:; d x |
2
ds ¡
∞; 06t ¡ 1:
12 Therefore Proposition A.4 immediately implies
Theorem 1.1. Let
k
t
:; d x = −
S
t
d x f
1 −t
S
t
− W
t
− 1
[S
t
; ∞[
x x
− W
t
1 − t
f
1 −t
x − W
t
d x; t
∈ [0; 1]: Then for 06t61 we have P
t
:; d x = P :; d x +
Z
t
k
s
:; d x dW
s
: Hence for our main example there is an integral representation of the process of
regular conditional densities of L. We denote k
t
:; d x also by D
t
P
t
+
:; d x; t ∈ [0; 1].
We now return to the general setting. It is clear that we just have to follow the ideas needed in the treatment of Example 1 to obtain a more general result, which we also
formulate in a weak version.
Theorem 1.2. Suppose that there exists an M
-valued process k
t
:; d x; t ∈ [0; 1] such
that for any t ∈ [0; 1]; f ∈ C
b
R we have E
Z
t
h[k
s
:; d x − k
s
:; d x]; f i
2
ds → 0
13
as → 0. Then for any t ∈ [0; 1]; f ∈ C
b
R hP
t
:; d x; f i = hP
:; d x; f i +
Z
t
hk
s
:; d x; f i dW
s
: If in addition
sup
f ∈ C
b
R; kfk61
E Z
t
hk
s
:; d x; f i
2
ds ¡ ∞;
14 then for any t
∈ [0; 1] P
t
:; d x = P :; d x +
Z
t
k
s
:; d x dW
s
:
2. The semimartingale property and relative entropy of the conditional laws
Kategori