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MATH503b
(SPRING 2010)
HOMEWORK #2
Due Wednesday, February 17
1) Let
H
be a complex inner product space, whose inner product is deﬁned by (
x,y
) =
x
y
, for all
x,y
∈
H
.
Let
k
x
k
:= (
x,x
)
1
/
2
. Prove the
CauchySchwartz inequality
:

(
x,y
)
 ≤ k
x
kk
y
k
,
∀
x,y
∈
H
.
Using the CauchySchwartz inequality to verify the
triangle inequality
:
k
x
+
y
k ≤ k
x
k
+
k
y
k
,
∀
x,y
∈
H
.
(Hint: To prove the CauchySchwartz inequality, you might want to ﬁrst choose a complex number
α
,
with

α

= 1, and
α
(
x,y
) =

(
x,y
)

, and then try to modify the argument in class.)
2) Assume
T
1
> T
2
, and let
M
be a martingale such that
E

M
T

2
<
∞
for all
T >
0. Then by Martingale
Representation Theorem, we can ﬁnd two processes
H
1
∈
L
2
T
1
(
B
) and
H
2
∈
L
2
T
2
(
B
), respectively, such
that it holds almost surely,
M
t
=
M
0
+
Z
t
0
H
1
s
dB
s
,
t
∈
[0
,T
1
];
and
M
t
=
M
0
+
Z
t
0
H
2
s
dB
s
,
t
∈
[0
,T
2
]
.
Argue that the representation is “consistent” in the sense that
H
1
t
=
H
2
t
, almost surely, for all
t
∈
[0
,T
2
].
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This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.
 Fall '09
 MAJIN
 Math

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