HW: 10
Course: M339D/M389D - Intro to Financial Math
Page: 1 of 4
University of Texas at Austin
HW Assignment 10
Asian options. Barrier options.
Provide your final answer only for the following problem(s):
Problem 10.1. (2 points)
In our usual notation, t
Math 131B-2
Homework 1 Solutions
Due: October, 4th, 2016
Exercise 1. Let H be a vector space over R and let h, i be an inner-product on H, i.e., H paired with h, i is a
real inner-product space. For x H, define
kxk = hx, xi1/2 ;
this is the norm on H inhe
Math 131B-2
Homework 5 Solutions
Due: November 8, 2016
Exercise 1. Use the Weierstrass approximation theorem to prove the following proposition.
Proposition 1. Let I = [0, 1] and suppose that f : I R is continuous. If
Z
xn f (x) dx = 0
I
for all n R N = c
Homework 2 Solutions
Math 131B-1
(3.29) Let (M, d) be a metric space, and
d (x, y) =
d(x, y)
.
1 + d(x, y)
d(x,x)
1+d(x,x)
0
1
We check that d is a metric.
For any x M , d (x, x) =
=
= 0.
(Positivity) For any x, y M such that x = y, d(x, y) > 0 and 1 +
Math 131B-2
Homework 7 Solutions
Due: December 2, 2016
Exercise 1. In this exercise, you will give another proof of Theorem 42 from the notes.
For each R, define
[] = cfw_ + 2k R : k Z
The collection of all such sets, called cosets, is denoted by
T = cfw_
Analysis II
Supplementary course notes for Math 131B
Evan Randles
These are the supplementary course notes for Math 131B. The notes pick up at the end of our treatment of
basic metric space topology from the courses official textbook, Analysis II by T. Ta
Math 131B
Fall 2016
Midterm Exam
October 28th, 2016
Name (Print):
UID:
This exam contains 5 pages (including this cover page) and 4 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your init
Basic Properties of Open and Closed Sets
David Jekel
January 16, 2017
The following notes cover material from Taos Analysis 1.2, especially Proposition 1.2.15.
1
Interior, Exterior, Boundary, Closure
Suppose (X, d) is a metric space and E X. The following
A Short Guide to Learning Mathematics
David Jekel
January 17, 2017
Preface and Disclaimers
This guide is primarily aimed at undergraduates in proof-based math classes,
but some of it may apply to other levels of math as well.
As you can gather from the ad
Math 131B-2
Homework 4
Due: October, 25th, 2016
This assignment is due on October 25th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises. In what follows, we assume the notation of the supplementary course
Math 131B-2
Homework 5
Due: November 8, 2016
Exercise 1. Consider the following definition (which is related, but not equivalent to the definition of adherent
point).
Definition 1. Let (X, d) be a metric space and E X. A point x X is said to be an accumul
Math 131B-2
Homework 3 Solutions
Due: October, 11th, 2016
Exercise 1. Consider the following definition.
Definition 1. Let (X, dX ) and (Y, dY ) be metric spaces and let f : D Y for some D X. Given a set E such
that D E X, a function g : E Y is said to be
Math 131B-2
Homework 2 Solutions
Due: October, 11th, 2016
Exercise 1. Let (X, d) be a metric space and Y X. A collection cfw_V I (where I is an index set) of open sets
in X is called an open cover of Y if
[
Y
V .
I
Any subcollection cfw_V K of cfw_V I (h
Math 131B-2
Homework 4 Solutions
Due: October, 25th, 2016
Exercise 1. Prove the following proposition.
(n)
Proposition 1. Let (X, d) be a metric space and let (f (n) )
)n=1 be sequences of functions from (X, d)
n=1 and (g
to R (equipped with the standard
Math 131B-2: Final Exam Study Sheet
Here is some general information about the final.
The final is Thursday, June 12th, 11:30-2:30, in Humanities 169 (our usual room).
I will have office hours Tuesday, June 10th, 1-3 p.m. and Wednesday, June 11th, 10 a.
Math 131B-2: Homework 9
Due: June 6, 2014
1. Read Tao Sections 16.3-5.
2. Prove Pythagoras Identity: If < f, g >= 0, then |f + g|22 = |f |22 + |g|22 .
3. Prove that the convolution f g of two continuous Z-periodic function is continuous. Hint:
You will ne
Math 131B-2: Homework 5
Due: May 5, 2014
1. Read Apostol Sections 4.16-17, 4.19-20, 9.1-5.
2. Do examples (a)-(d) of problem 4.11 in Apostol. [You do not have to do the proof at the
beginning.]
3. Do problems 4.28, 4.37, 4.49, 4.52, 4.54 in Apostol.
[Note
Math 131B-2: Homework 3
Due: April 21, 2014
1. Read sections 4.1-5, 4.8-9 in Apostol.
2. Do problems 3.38, 3.42, 3.17, 3.20, 3.22, 4.7, 4.8, and 4.9 in Apostol.
Notice that problem 3.38 justifies our assertion that compactness is an absolute, not relative
Lecture: 6
Course: M339D/M389D - Intro to Financial Math
Page: 1 of 5
University of Texas at Austin
Lecture 6
An introduction to European put options. Moneyness.
6.1. Put options. A put option gives the owner the right but not the obligation to
sell the u
Math 131B
Spring 2017
UCLA
Homework 6
Due on Friday, May 12th
Exercises from the text:
Section 3.1: 2ab
Hint for 3.1.2ab: This is extremely similar to the proof of Theorem 2.1.4ab, which we
did in class.
Section 3.2: 2
P
(n) does not converge uniformly
Fo
Math 131B
Spring 2017
UCLA
Homework 4
Due on Friday, April 28th
Exercises from the text:
Section 1.5: 3, 4, 5
For 1.5.3: When the book references Theorem 9.1.24, the thing to use is the BolzanoWeierstrass Theorem from Math 131A, which we recalled in lectu
Math 131B
Spring 2017
UCLA
Homework 3
Due on Friday, April 21st
Exercises from the text:
Section 1.1: 12
Hint for 1.1.12: Use the inequalities (1.1) and (1.2) (on pages 4 and 5, respectively)
which you proved on Homework 1.
Section 1.3: 1
Hint for 1.3.1:
Math 131B
Spring 2017
UCLA
Homework 7
Due on Friday, May 19th
Exercises from the text:
Section 3.4: 1, 2
Section 3.6: 1
Additional problems:
(1) Prove the Weierstrass M -test as stated here:
Theorem (Weierstrass M -test). Let (X, d) be a metric space, and
Math 131B
Spring 2017
UCLA
Homework 2
Due on Friday, April 14th
Exercises from the text:
Section 1.1: 13, 15, 16
Section 1.2: 3fgh, 4
Additional problems:
(1) Let (X, d) be a metric space. The triangle inequality
d(x, z) d(x, y) + d(y, z)
for all
x, y, z
Math 131B
Spring 2017
UCLA
Homework 1
Due on Friday, April 7th
Exercises from the text:
Section 1.1: 3, 8, 9, 10
Hint for 1.1.8: You may use (without proof) any part of Exercise 1.1.5.
Additional problem:
(1) Let (X, d) be a metric space.
(a) Show that fo
COUNTABLE PRODUCTS
ELENA GUREVICH
Abstract. In this paper, we extend our study to countably infinite products of topological
spaces.
1. The Cantor Set
Let us constract a very curios (but usefull) set known as the Cantor Set. Consider the
closed unit inter
Math 112
Solutions for Problem Set 2
Spring, 2013
Professor Hopkins
1. (Rudin, Ch 1, #6). Fix b > 1.
(a)
If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that
(bm )1/n = (bp )1/q .
Hence it makes sense to define br = (bm )1/n .
(b)
Prove
Math 140B
HW 7, May 20 Chapter 7, Page 167-168: #13(a), 15, 16, 18.
#13. Assume that ffn g is a sequence of monotonically increasing functions on R with 0
fn (x) 1 for all x and all n:
(a) Prove that there is a function f and a sequence fnk g such that
f