ACTSC/STAT 446/846
Tutorial #1
January 21, 2009
Let
W
=
{
W
t
, t
≥
0
}
be a standard Brownian motion.
1. Let
Z
1
, Z
2
, . . .
be positive iid random variables with mean
1
. Define
X
0
=
x
and
X
n
=
Q
n
j
=1
Z
j
for all
n
≥
1
. Show that
X
=
{
X
n
, n
≥
0
}
is a martingale with respect to the filtration generated
by the
Z
n
’s and find its mean.
2. The price of a stock at time
n
is modelled by
S
n
=
S
0
e
ε
1
+
···
+
ε
n
,
where
S
0
is a positive constant and where the
ε
k
’s are i.i.d. random variables with common
distribution
(
a
with probability
q
;
b
with probability
1

q
.
The continuously compounded riskfree rate is given by
r
.
Find the value of
q
for which the
discounted price
D
n
=
e

rn
S
n
is a martingale with respect to
F
=
{F
n
}
n
≥
0
, the filtration
generated by the
ε
k
’s. If you need any restrictions on
a, b
and
r
, state and justify them.
3. Let
B
=
{
B
t
, t
≥
0
}
be the Brownian motion with drift
μ
and diffusion coefficient
σ
obtained as
a linear transformation of
W
.
(a) Prove that
C
ov
(
W
s
, W
t
) = min
{
s, t
}
;
(b) Compute the covariance of
W
s
and
B
t
;
(c) Compute
E
e
B
t
.
4. Let
X
= (
X
t
)
0
≤
t
≤
1
be the socalled Brownian bridge, that is, for
0
≤
t
≤
1
, define
X
t
=
W
t

tW
1
.
Compute the mean, variance and covariance functions of
X
.
5. Assume that
S
=
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 Spring '09
 idk..
 Normal Distribution, Variance, Brownian Motion, Probability theory, φ, standard Brownian motion

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