Stochastic Processes and Financial Mathematics
(part two)
11.5 Exercises on Chapter 11
In all the following questions,
On Brownian motion
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11.1 Consider the process
, for , where and are deterministic constants.-
(a) Find the mean and variance of
. -
(b) Let
. What is the distribution of ? -
(c) Is
a random continuous function? -
(d) Is
a Brownian motion?
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11.2 Let
. Use the properties of Brownian motion to show that . -
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(a) Show that
for all . (Hint: Integrate by parts!) -
(b) Deduce that
and . -
(c) Write down
for any . -
(d) Show that
for all .
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11.6 Show that the following processes are martingales.
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(a)
where is a deterministic constant. -
(b)
.
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11.7 Fix
and for each let be such that and as . (For example: ).-
(a) Show that
and . -
(b) Show that
as . -
(c) Set
. Show that and that as .
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