is measurable on the product space [0, ), where is the probability space on which cfw_Bt : t 0 is dened.
In order to avoid some annoying hassle with measurability, we assume without losing the generality that
cfw_Bt : t 0 is dened on the canonical space =
Since cfw_ei : i I is an orthonormal basis of H, the net (PJ ( f )J D converges to f in H. In particular, it is a
Cauchy net in H. Then by utilizing the isometry (1.4), we nd that
X J ( f ) X J ( f ) L2 2 (P) = X J \J ( f ) X J \J ( f ) L2 2 (P)
= X J \J
Stochastic Calculus HW #1
Sangchul Lee
January 22, 2016
1
Solutions
Exercise 1.1. As a review, prove the (what the textbook calls) Doob-Dynkin lemma: If X is an R-valued
random variable and Y is (X )-measurable, then there exists a Borel measurable f : R
Solutions for Problem Set #5
due October 17, 2003
Dustin Cartwright and Dylan Thurston
1 (B&N 6.5) Suppose an analytic function f agrees with tan x, 0 x
1. Show that f (z) = i has no solution. Could f be entire? (not graded)
Since f (z) agrees with tan z
Solutions for Problem Set #4
due October 10, 2003
Dustin Cartwright
R
(B&N 4.3) Evaluate C f where f (z) = 1/z as in Example 2, and C is given
by z(t) = sin t + i cos t, 0 t 2. Why is the result different from that of
Example 2? (not graded)
We can substi
Mathematics 113
Fall 2001
Homework Assignment # 6
Due Thursday, November 15
1. a. If Log denotes the principal branch of the logarithm, simplify the
expressions
eLog(i) , Log(i), iLog(1) , (1 + i)Log(1+i) ,
where z w = ewLogz .
b. Find the Taylor expansio
Homework #3Part 1 Solutions
Complex Variables
Instructor: Irina Nenciu
R
Problem 4.9. Evaluate C (z i) dz where C is the parabolic segment:
z(t) = t + it2 ,
1 t 1 .
2
Solution. (a) The function F (z) = z2 iz is entire and F 0 (z) = f (z).
Therefore,
Z
(1
WRITTEN HOMEWORK #4, DUE 4/23/2012 AT 4PM
You can turn this in during class on April 23 or at my office (Kemeny 316) by 4pm.
Please make sure your homework assignment is stapled, if necessary, before handing
it in. (In particular, there is no guarantee th
Mathematics 113
Fall 2001
Homework Assignment # 5
Due Thursday, October 25
1
z
3
1. Find power series expansions for f (z) = (z+1)
2 , g(z) = e , and h(z) = z
centered at z = 1 + i. What are the radii of convergence of these power
series?
2. a. Find all c
Do the following:
(1) Show that f t (x) := mincfw_x, t lies in H for each t 0.
(2) Setting Bt := X ( f t ) (for X ( f ) as above) denes a process cfw_Bt : t 0 whose nite dimensional
distributions are those of the standard Brownian motion.
Proof. (1) If t
Exercise 1.4. Let cfw_Bt : t 0 be the d-dimensional Brownian motion and let U be an orthogonal matrix.
Show that also U Bt is a d-dimensional Brownian motion.
Proof. We check that U Bt satises all the requirements for Brownian motion. First, the following
If x E, then there exists n and N such that f k (x) Y L + n1 for all k N. Using this, we nd that
cfw_Yk Y L + n1 for all k N .
cfw_X E
n 1 N 1
But since Yk Y , the right-hand side is a P-null set.
Summarizing, we have Y = Y 1 cfw_X E = f (X )1 cfw_X
United Arab Emirates University
College of Sciences
Department of Mathematical Sciences
HOMEWORK 1 cfw_ SOLUTION
Section 1 cfw_ Section 5
Complex Analysis I
MATH 315 SECTION 01 CRN 23516
9:30 cfw_ 10:45 on Monday & Wednesday
Due Date: Monday, September 13
Complex Analysis for Applications, Math 132/1,
Home Work Solutions-II
Masamichi Takesaki
Page 148, Problem 1.
0
Page 129, Problem 2. If two contours 0 and 1 are respectively shrunkable to
single points in a domain D, then they are continuously deformable
Solutions for Problem Set #1
due September 19, 2003
Dustin Cartwright
repeated application of our result from part b,
proves that z n = z n
P (z) = an z n + . . . + a0
= an z n + . . . + a0 (by part a)
= an z n + . . . + a0 (by part b)
(B&N 1.4) Prove the
Solutions for Problem Set #2
due September 26, 2003
Dustin Cartwright
(B&N 2.2) By comparing coefficients or by use of the Cauchy-Riemann
equations, determine which of the following polynomials are analytic. (5
points each)
a. P (x + iy) = x3 3xy 2 x + i(
MATH421 DISCUSSION LESSON PLAN WEEK 7
CHRIS JANJIGIAN
This week, I am going to focus on sup, inf, lim sup and lim inf . We probably
will not be able to cover all of this in discussion, but the properties and results
here are a good way to get comfortable
MA671 Assignment #2
Morgan Schreffler
Page 1 of ?
X
Exercise 1 Assume the power series f (z) =
Cn (z z0 )n converges for 0 < |z z0 | < R.
n=0
Show that f is continuous at z0 ; that is,
lim f (z) = f (z0 ).
X
Proof. Recall that if a power series f (z) =
Cn
Math 312
Assignment 4 Key
Section 1
Spring 2008
1. 4.7. Give a direct proof of the Lemma 4.14. That is, given any rectangle with
vertices (a, c), (b, c), (b, d), and (a, d), parametrize the boundary and verify directly
that
dz = 0 =
z dz.
Show your work.
Homework #2 Solutions
Complex Variables
Instructor: Irina Nenciu
Problem 2.3. Show that no nonconstant analytic polynomial can
take imaginary values only.
Solution. Let P be an analytic polynomial taking only imaginary
values. Then we know that
(1)
Py = i
lim sup
Definition 1. Suppose the sequence cfw_an is bounded.
lim sup an = lim (supcfw_ak : k n).
n
n
Theorem 1. There is a unique number a such that:
Given any > 0 there are infinitely many n such that an > a , but only finitely many n such that
an > a
Challenge Problem
October 4, 2016
1
Problem 1
Knowing nothing about the exponential function show that ex+y = ex ey
Let x, y R
n!
The binomial coefficient gives k!(nk)!
= nk
n
X
n nk n
n
x
y
and the so we get (x + y) =
k
k=0
X
X
xn
yn
Starting with ex ey
Function Theory
a Lecture by Prof. Imamoglu
Notes by Ruben Andrist, Thomas Rast and Simon Wood
Warning: We are sure there are lots of mistakes in these notes. Use at your
own risk! Corrections and other feedback would be greatly appreciated and
can be sen
Let () = (X ) 1 () be the normalized measure on X .
Generate a sequence (Un ) of i.i.d. random variables having the law , i.e., P(Un ) = ().
law
Let N = Poisson(X ) be independent of (Un ).
Then we dene cfw_X ( A) : A F as
X ( A) =
N
1 A (Uk ).
k=1
By