ACTSC/STAT 446/846
Assignment #1
Due date: January 23, 2009
Note:
Recall that, when handing in your assignment, you are requested to use a cover page showing only your UWID
number and your section (846 students: also write “846” on the cover page) and to write your full name on the next
page.
Assignments must be handed in during the TAs office hours
before or on the due date (see the Calendar on UWACE
for details).
Let
W
=
{
W
t
}
t
≥
0
be a standard Brownian motion.
1. Problems 18.9 and 20.9 in McDonald’s book
1
.
2. Let
0
≤
t
1
< t
2
.
(a) Write the probability density functions of
W
t
2

W
t
1
and
W
t
1

W
t
2
.
(b) Define
Y
t
=
αt
+
βW
t
, where
α
and
β
are constants. Write the probability density function
of
Y
t
.
(c) If
W
t
1
=

π
, write the (conditional) probability density function of
W
t
2
.
3. Let the price at time
t
of RIM stock be given by
S
t
=
S
0
exp
{
μt
+
σW
t
}
,
where
μ
and
σ
are constants.
(a) Find the SDE for which
S
t
is a solution. If
σ
=
S
0
= 1
, for which value(s) of
μ
is
t
7→
E
[
S
t
]
a constant function.
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 Spring '09
 idk..
 Probability theory, probability density function, Ito’s formula, Deﬁne Yt

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