Fall 2016 MATH 526
Hw 5
Show all your work for full credit. Due: December 8, in class. All four questions are of the same value.
1. In discrete time, assuming the interest rate is zero (which is what we currently observe), a very general model for
the sto
Math 472, F 2016, Homework 4
Due Friday Nov 11 at 4 pm
Include a cover page
Always clearly label all plots (title, x-label, y-label and legend)
Use the subplot command from MATLAB when comparing 2 or
more plots to make comparisons easier and to save pa
526 Stochastic Processes, Fall 2016
Homework 1
Due: Monday, Oct 3, 2016 in class
1. (20 pts) A taxicab driver moves between airport A and two hotels B and C according
to the following rules. If he is at the airport, he will be at hotel B next with probabi
Math 472, F 2016 Homework 2
Due Friday Sept 30 at 4 pm
Include a cover page
Always clearly label all plots (title, x-label, y-label and legend)
Use the subplot command from MATLAB when comparing 2 or
more plots to make comparisons easier and to save pa
Math 472, F 2016, Homework 3
Due Fri Oct 14 at 4 pm
Include a cover page
Always clearly label all plots (title, x-label, y-label and legend)
Use the subplot command from MATLAB when comparing 2 or
more plots to make comparisons easier and to save paper
Math 472, F 2016, Homework 1
Due Thu Sep 15 at the beginning of class
Include a cover page
Always clearly label all plots (title, x-label, y-label and legend)
Use the subplot command from MATLAB when comparing 2 or
more plots to make comparisons easier
573 Financial Mathematics I, Fall 2016.
Homework 2
Due: Tuesday, Oct 19, 2016, NO LATER than 5:00pm.
The maximum number of points you can receive for this homework is 30.
1. (3 pts) Consider a trinomial model which has one riskless and one risky asset. Th
526 Stochastic Processes, Winter 2015.
The suggested problems below are for practicing ONLY. They are NOT part of your assignment and will not be
graded.
* Quizzes and exams may contain similar problems. *
Suggested problems on Brownian motion.
Find an e
Z
=6
e5v
0
Thus, we conclude
ET =
e2v
1
2
4
dv = 1
3
7
=
10
10
59
1 1 1 1 7
+ +
=
hours
3 2 5 2 10
60
(3 pts)
2. (3 pts) Suppose that the number of calls per hour to an answering service follows a Poisson process with rate
10 per hour. Suppose that each
Solution to Quiz 4
526-W15
1. True or False? (Each .5 pts)
a) If X and Y are normal random variables, then, X + Y has normal distribution.
b) If (X, Y ) is a Gaussian vector, with the covariance matrix
!
2 2
2 3
then, it has a probability density function
Answer Key to Quiz 1
526-W15
1. (2 pts) Suppose that the probability it rains today is 0.3 if neither of the last two days
was rainy, but 0.6 if at least one of the last two days was rainy. Let the weather on day n,
Wn , be R for rain, or S for sun.
(a) I
Answer Key to Quiz 3
526-W15
1. Consider a cat chasing a mouse. At the jump times of a Poisson process with rate 2, the
mouse moves 1 or 4 steps ahead, with probabilities 2/3 and 1/3, respectively. At the same
times, the cat moves 3 steps, if the mouse is
526 Stochastic Processes, Winter 2015.
The suggested problems below are for practicing ONLY. They are NOT part of your
assignment and will not be graded.
* Quizzes and exams may contain similar problems. *
All the problems are from Essentials of Stochasti
526 Stochastic Processes, Winter 2015.
Solution to some of the suggested problems on Martingales.
This is for practicing ONLY, it is NOT a part of your assignment and will not be graded. Hwks, quizzes, and exams
may contain similar problems.
From the Ess
526 Stochastic Processes, Winter 2015.
The suggested problems below are for practicing ONLY. They are NOT part of your
assignment and will not be graded.
* Quizzes and exams may contain similar problems. *
All the problems are from Essentials of Stochasti
526 Stochastic Processes, Winter 2015.
Solution to Homework 1
1. (2 pts) Find the stationary distribution of a Markov chain with the following transition matrix:
1
2
3
4
1
2
3
4
0 2/3 0 1/3
1/3 0 2/3 0
0 1/6 0 5/6
2/5 0 3/5 0
Solution
P
To find the statio
526 Stochastic Processes, Winter 2015.
The suggested problems below are for practicing ONLY. They are NOT part of your
assignment and will not be graded.
* Quizzes and exams may contain similar problems. *
All the numbered problems are from Essentials of
526 Stochastic Processes, Winter 2015.
Homework 2
The maximum number of points you can receive for this homework is 12.
Solutions
From the Essentials of Stochastic Processes, 2nd ed. (Durrett):
(2pts) 1.57
Solution 1: The problem is the same as the gamb
526 Stochastic Processes, Fall 2016
Homework 3
Due: Wednesday, Nov 9, 2016 in class
1. (15 pts) Assume that a company stock price (Xn ) is a random walk reflected at 1 and
at 4: i.e. it is an MC, with S = cfw_1, 2, 3, 4 and transition matrix:
1
2
3
4
1
2
573 Financial Mathematics I, Fall 2016.
Lecturer: Sergey Nadtochiy.
Lecture 4. Optimal investment in single-period models.
1
Mean-variance portfolio optimization
We work here with a one-period model with d risky assets. The corresponding price vector at t
Chapter 12
Multiple Regression
Learn.
To use Multiple Regression
Analysis to predict a response
variable using more than one
explanatory variable.
Agresti/Franklin Statistics, 1 of 141
Section 12.1
How Can We Use Several
Variables to Predict a
Response?
Math 472 Numerical Methods with
Financial Applications
Instructor: Christian Keller, 4843 EH, ckell@umich.edu
Time and location: TuTh 8:30am-10:00am, 1505 CCL
Office hours: Tu 10:00am-12:00pm, Th 10:00am-11:00am, or by appointment
Course description: This
Answer key to Quiz 3.
526-F12
Problem. Consider a customer service office with two representatives, each one requires
an independent exponentially distributed period of time with mean 10 minutes to set up a
customer. Customers call the office at times of
Math 472 - Homework 2 (due Thursday, September 22,
2016): Note that [S] refers to our textbook.
(1) Exercise 0.3.15 (a) in [S](10 points). For this problem, you can
use the binary representation of (0.3)10 from Lecture 1.
(2) Exercise 0.4.2 in [S] (5 poin
Fall 2016 MATH 526
Hw 2-Solutions
1. Solution to 2.15:
Lets state and prove the following result first:
Suppose X1 exp(1 ) and X2 exp(2 ). Then P(X1 < X2 ) =
1
1 +2 .
1
1 +2
and E(X1 |X1 < X2 ) =
The first result was explained in class, as for the conditi
Solution to Practice Midterm
526-F16
1. A warehouse has a capacity to hold 2 items. If the warehouse is neither full nor empty, the number
of items in the warehouse changes whenever a new item is produced or an item is sold. Suppose that, on
any given day
Midterm-Math 526, Fall 2016
1. (25 points)
Roll a fair (six-sided) die repeatedly and let Y1 , Y2 , . . . be the resulting outcomes (numbers
from 1 to 6). Let Xn = |cfw_Y1 , Y2 , . . . , Yn | be the number of distinct values we have seen in the
first n ro
Fall 2016 MATH 526
Hw 2-Solutions
1. Let X be a Markov chain with state space E = cfw_a, b and transition matrix
0.4 0.6
P =
,
1
0
and suppose that a reward of g(i, j) units is received for every jump from i to j where
3 2
g=
.
1 1
Find
n
1 X
g(Xm , Xm+1
526 Stochastic Processes, Solutions to Practice Problems.
1. Our state space is S = cfw_P, N where P = positive daily return and N for negative daily
return. The transition matrix is
P N
P 0.8 0.2
N 0.3 0.7
(a) Let q = (0.6, 0.4) be the distribution at t
526 Stochastic Processes, Fall 2016.
Homework 1
Due 29th of September. Will be collected in the beginning of class. Late submissions
will not be accepted.
1. ( 10 pts)
The Smiths receive the paper every morning and place it on a pile after reading it.
Eac
Solution to Quiz 4
526-F16
1. Cars arrive at a three-pump gas station at the rate of 10 cars per hour. However, a
car will go to another station if there are at least 3 cars being served at the station. Suppose
that the service time for each pump is expon
Solutions to Quiz 3
526-F16
(1) Solution 1: The problem is the same as the gamblers ruin problem with probability
of win p = 2/3. You can use the formula in the book to get:
Px (VN < V0 ) =
with N = 4, x = 1 and
1p
p
x
1 ( 1p
p )
N
1 ( 1p
p )
,
= 1/2. The
Fall 2016 MATH 526
Solutions to the Self Assessment Test
1. Consider three points, Brighton, Ann Arbor, and the Airport. There are two roads available between
Brighton and Ann Arbor. There are also two roads between Ann Arbor and the Airport. Each of the
Answer Key Quiz 1
526-F16
1. 1. In a retail shop, they check the warehouse each morning at 8am. If they are out of
a specific product, e.g. product A, they order three and receive them next morning, right
before 8am. Any available product can be sold duri
Math 526 Fall 2016 Homework #4
Due Tuesday November, 22, 10.10 a.m. Bring your Hw to the class. Last problem is 20
points, the others are 16 each.
Problem 1
A physical device can be in three states: A, B, C. The device operates as follows (all time units