Numerical Analysis I, Lecture 6, Trigonometric
interpolation
Wujun Zhang
September 23, 2016
Content
interpolation by trigonometric functions
the finite Fourier transform
fast Fourier transform
materials from the books [A: 3.8], [KC: 6.12, 6.13], [QSS:
Numerical Analysis I, Lecture 7, Piecewise
polynomial approximation in two dimensions
Wujun Zhang
September 27, 2016
Content
piecewise polynomials in two dimensions
materials from book [KC: 6.10], [QSS: 8.6]
1
Piecewise polynomials in two dimensions
We
Numerical Analysis I, Lecture 3, polynomial
interpolation
Wujun Zhang
September 14, 2016
Content
Interpolation of moments
Piecewise polynomial interpolation
materials from the books [A: 3.5], [KC: 6.1], [QSS: 8.1], [SB: 2.1.4]
1
Interpolation of Moment
Numerical Analysis I, Lecture 1, polynomial
interpolation
Wujun Zhang
September 14, 2016
Content
Lagrangian and Newton forms of the interpolating polynomial
materials from the book [A: 4.1, 3.1, 3.2], [KC: 6.1, 6.2], [QSS: 8.1, 8.2],
[SB: 2.1.1, 2.1.3].
MATH 573 ASSIGNMENT 1 SOLUTIONS
1. Plots on later pages.
Problem 1a
n max |e(x)| approx location
4
.44
4
8
1
4.5
16
14
4.8
32
4500
4.9
Problem 1b
n max |e(x)| approx location
4
.45
4
8
.38
4.5
16
.068
4.6
32 3 105
4.8
2. Plots on later pages.
n max |e(x)|
Math Finance I PS 1
Yin Haobo/165000142
September 8, 2015
1. Proof:
Since X1 = (1 + r)(X0 D0 S0 ) + D0 S1 and suppose there exists arbitrage, which means X0 = 0,
X1 0, and either X1 (H) or X1 (T ) is strictly larger than 0.
So if X1 (H) = D0 S0 (1 + r)D0
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Math 622
Lecture Notes I
Spring 2013
Jump Process Models: Part I
I. Overview of price modeling in continuous time
Let cfw_F(t); t 0 be a ltration modeling the accumulation of market information
available to investors as time progresses. A simple paradigm
Mathematical Finance II
Spring 2013
Lecture Notes for Lectures 9 and 10
These notes review setting up of multi-asset models and the mathematical techniques for their analysis. As an example, a model of a market with a tradeable foreign
currency is given.
642:622; Notes for Lecture 7; Spring 2013
Asian Options
I. Asian Options: Denitions and Examples.
Let S(t) denote the price of an underlying asset. Its average price over the time
interval [0, T ] is
1 T
Save (T ) =
S(u) du.
T 0
Options whose payo depends
Math 622; Lecture Notes 7; American Options; Spring 2013
I. Denitions and Examples.
American options are options which the owner may exercise at any time between
date of purchase (t = 0) and expiry (t = T ).
Consider an underlying asset with price cfw_S(t
Math 622
Notes to Lecture 4, Part I
Spring 2013
I. Useful review and denitions for the theory of stopping times.
(a) -algebras and measurability.
A collection G, of subsets of a set , is a -algebra if it is closed under the
operations of complementation,
Math 622
Notes to Lecture 5, Part II
Spring 2013
4. Pricing a lookback option for the Black-Scholes price.
Lookback options are options whose payos depend on the maximum or minimum
of the underlying assets price over the lifetime of the option. For exampl
Math 622
Notes to Lecture 6
Spring 2013
1. An extension of Itos rule.
Let cfw_S(t) represent an asset price and let Y (t) := maxcfw_S(u); 0 u t denote
its running maximum. This notation will be used throughout the lecture.
When the process S is continuous
Mathematical Finance II
Spring 2013
Lecture Notes for Lecture 5
1
1. The reection principle for Brownian motion.
Let W be a Brownian motion with a ltration cfw_F(t); t 0, and let be a
stopping time. Dene
B (t) =
W (t),
if t ;
W ( ) [W (t) W ( )], if t >
Math 622
Notes to Lecture 4, Part II
Spring 2013
An addendum to Part I of Lecture 4. We state without proof an extension
of Theorem 4 of Part I.
t
Theorem 1 Let X(t) = X(0) +
t
(s) ds +
0
(s) dW (s). Assume that b > X(0)
0
and let Tb be the rst time X hit
Notes on the Dunford-Pettis compactness criterion The Dunford-Pettis compactness criterion implies that uniform integrability is a necessary and sufficient condition for weak sequential compactness of a family of integrable random variables. The theorem a
NOTES AND EXERCISES, LECTURE 1, MATH 642:592, Spring 2008
1. Probability space, random variables and expectation.
We summarize the formal mathematical setting of the course:
(1) All analysis takes place in a probability space. This is a triple (, F , P),
NOTES AND EXERCISES, MARTINGALES, MATH 642:592, Spring 2008
The material covered in lectures 2, 3, and 4 on conditional expectation and discrete time
martingales is standard and may be found in RW, Volume I. Therefore, we mainly summarize
here and give in
BROWNIAN MOTION, MATH 642:592, Spring 2008
1. Stochastic Processes in Continuous Time.
A stochastic process in continuous time is a family X = cfw_X (t); t 0 of random variables on a
xed probability space, indexed by all real t 0. Given , the function t X
BROWNIAN MOTION-continued, MATH 642:592, Spring 2008
1. Lvys construction of Brownian motion. This is nicely covered in RW, Chapter 1, section 6.
e
The approach we took in lecture was dierent only in the way we expressed it. (The particular
way we did the
Lvy Processes, Math 642:592, Spring 2008
e
1. L`vy Processes and Poisson processes.
e
Denition: A function f : [0, ) R is c`dl`g if it is right-continuous at all t [0, ) and if the
aa
limit from the left
f (t) = lim s t, s < tf (s)
exists and is nite for
Introduction to stochastic integration: Math 642:592, Spring 2008
I. Bounded variation functions and Lebesgue-Stieltjes integrals.
As a preliminary to the theory of stochastic integration, we recall the theory of Lebesgue-Stieltes
integrals and its relati
Filtrations, Stopping times, and some applications: Math 642:592, Spring 2008
1. Filtrations for continuous time processes.
A ltration F = cfw_Ft t0 in a measure space (, F ) is an increasing family of sub- -algebras of
F . Here t ranges through all non-n
2
MATH 574 LECTURE NOTES
1. Solution of linear systems of equations by direct methods
We consider the problem of nding a vector x = (x1 , x2 , , xn )T satisfying the linear
system of equations Ax = b, where A = (aij ) is the square matrix
a11
a21
A=
an
6
MATH 574 LECTURE NOTES
1.2. Choleski decomposition. We next look at a simplication of the LU decomposition
algorithm in the case when A is symmetric, i.e., we seek a factorization of the form A = LLT ,
known as Choleski decomposition.
We rst observe tha
MATH 574 LECTURE NOTES
9
2.1. Perturbation theory for linear systems of equations. We next try to understand
for what types of matrices, small changes in the entries of the matrix or small changes in the
right hand side have the potential to produce large