Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Mid-Term Exam 1 Suggested Answers
Closed book. You have one hour to complete the exam. The exam consists
of xxx numbered pages.
Grade value: 20 % of 200 points = 40 points. Problems AD.
Ad
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 1
Grade value: 10 % of 200 points = 20 points. Every question gives two
points, except where otherwise indicated.
Due Wednesday October 7. Please post your answers in Coursewor
Solutions to Selected Exercises in Chapter 9, Fall 2015
Lars Tyge Nielsen
Exercise 9.1 and Exercise 9.2
Versions of these exercises are in Problem Set 3, and he solutions will be
included in the suggested answers to that problem set.
Exercise 9.3
Define a
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 1
Grade value: 10 % of 200 points = 20 points. Every question gives two
points, except where otherwise indicated.
Due Wednesday October 7. Please post your answers in Coursewor
Solutions to Exercises in Chapter 8, Fall 2015
Lars Tyge Nielsen
Exercise 8.1
Assume that
a(t) = 1cfw_0 (t) +
n1
X
i 1(ti ,ti+1 ] (t)
i=0
and
b(t) = 1cfw_0(t) +
n1
X
i 1(ti ,ti+1 ] (t)
i=0
where 0 = t0 < < tn = t, , F0, and the random variables i , i are
Solutions to Selected Exercises, Fall 2015
Lars Tyge Nielsen
Exercise 8.1
The payoff to the option is
Y
=
S(T )
(
1
0
if
M (T )
X
(T )
M
S(T )
otherwise
This is the payoff to a cash-or-nothing put option on asset zero, with trigger
price M(T )/X. Write d2
Appendix A
Suggested Solutions to
Exercises
c 2015 Lars Tyge Nielsen All Rights Reserved Do not post or circulate
A.1
Solutions for Chapter 1
Exercise 1.1
Proof of Proposition 1.1.
1: Since An F for all n, Acn F for all n, and so
quently,
"
#c
\
[
An =
Ac
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 3 - Suggested Answers
Grade value: 10 % of 200 points = 20 points. Problems AD.
Due Monday November 16. Please post your answers in Courseworks.
Problem A [4 points]
Let W be a
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Mid-Term Exam 2 Suggested Answers
Closed book. You have one hour to complete the exam. The exam consists
of 4 numbered pages.
Grade value: 20 % of 200 points = 40 points. Problems AI. That
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 2, Suggested Answers
Grade value: 10 % of 200 points = 20 points. Three pages. Five problems,
AE. Max four points per problem.
Due Wednesday October 21. Please post your answer
Appendix A
Suggested Solutions to
Exercises
c 2015 Lars Tyge Nielsen All Rights Reserved Do not post or circulate
A.1
Solutions for Chapter 1
Exercise 1.1
Proof of Proposition 1.1.
1: Since An F for all n, Acn F for all n, and so
quently,
"
#c
\
[
An =
Ac
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 4
Grade value: 10 % of 200 points = 20 points. Problems AE.
Due Wednesday December 9. Please post your answers in Courseworks.
A [2 points] Exercise 9.6 from the Lecture Notes
44
APPENDIX D.
SUGGESTED SOLUTIONS TO EXERCISES
Exercise 9.5
(1)
By Exercise 9.3,
1
dWk = kWk1 dW + k(k 1)Wk2 dt
2
In integral form,
Z
k
W (t) =
t
0
Z
kW
k1
dW +
t
0
1
k(k 1)Wk2 ds
2
k1 is in H 2 by ExerThe stochastic integral
is a martingale because kW
c
Stochastic Processes and Applications, Fall 2015
Lars Tyge Nielsen
Problem Set 3
Grade value: 10 % of 200 points = 20 points. Problems AD.
Due Monday November 16. Please post your answers in Courseworks.
Problem A [4 points]
Let W be a one-dimensional Wie
G6501-001, Fall 2015: Homework set 6
Due Wednesday December 9
1. Page 526, exercise 11.4. (Please add one more assumption that N1 and N2 are independent.)
2. Page 526, exercise 11.5.
3. Let X and Y be continuous local martingales. Show that
E(X)E(Y ) = E(
G6501-001, Fall 2015: Homework set 4
Due Wednesday November 11
1. Page 191, exercise 4.6
2. Page 197, exercise 4.13
3. (Integration by Parts) Let W be a Brownian motion. Use Itos formula to prove that if h is a deterministic, continuously differentiable f
G6501 Stochastic Processes: Lecture 26
Yuchong Zhang
Wednesday December 9, 2015
Yuchong Zhang
G6501 Stochastic Processes: Lecture 26
Plan
I
Summary
I
Practice problems
I
Q&A
Yuchong Zhang
G6501 Stochastic Processes: Lecture 26
Main topics discussed
I
BM,
G6501-001, Fall 2015: Homework set 5
Due Wednesday November 25
1. Page 282, exercise 6.1
2. The four pictures below show generic sample paths of the solutions X to four different SDEs. Match
the sample paths with the SDEs. Explain your choices.
(a) dX = d
G6501-001, Fall 2015: Homework set 3
Due Monday October 19
1. For each of the following statements, indicate whether it is true or false. If the statement is true, explain
why. If the statement is false, also explain why.R A correct explanation is require
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M f
q) ¥$d3e. 'frke 18Q&VWH£ Vafﬁﬁiom {S a Yam90w\
“ ”' Mb€ﬁ
Vcwmlola “We voumcmce IS 6x W
a <M‘§)JC+<T\«/t 7 11M
‘¥> :mke, §3trgm~e .TWW {WPM6%I
L5 01 INDIA- ﬂgejcd’Q/‘e LVALﬁbm \ g} >0 MC §D> O
C) Fade?“ 'Trg/e (1C W hom'ﬂﬁmc/Om/ [Qt/(f- 140+
m agmm