APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.12, SOLUTION
Peter Lakner
(a) Let q1 + q2 = a and q3 + q4 = 1 a. Since 5 is the average of 2 and 8, we have a = 12 .
Also, 8 is the average of 6 and 10 so q3 = q4 = 14 . In order to determine q1 and q2

Lecture 3
4.5 Asymptotic Approximations for Binomial Random Variable
A
A
P ( A) p, P ( A) q, p q 1
n
Pn (k ) Pcfw_ A occurs k times in n trials p k q nk
k
k
n
Pcfw_ A occurs k1 to k2 times in n trials p k q nk
k k k
The Normal Approximation (DeMoivre-La

1EL6303
HW 4
Fall 2016
1. Find and draw FY ( y ) and fY ( y ) in terms of FX ( x) and f X ( x) .
2, for x 0
0, for 0 x 1
y g ( x)
x 1, for 1 x 2
1, for x 2
fX ( x)
x
(Video is required.)
2. The random variable X is uniform in the interval (1, 1). Fin

EL6303
HW 1 Solution
1. For any arbitrary events
A, B, C
with
P( A U B)| C ) 1 P( A U B UC )
P(C )
Solution:
Theorem:
Fall 2016
P(C ) 0
, prove or disprove that
.
.
A B P( A) P(B)
C is a subset of A U B UC P( A U B UC ) P(C)
P( A U B)C ) P( A U B UC ) P(

Lecture 4
Example Y g ( X ) .
y g ( x)
1
1
1
1
x
Find FY ( y ) and fY ( y ) in terms of FX ( x) and f X ( x) .
1, y 1
Solution: We see that | Y | 1. Thus FY ( y )
0, y 1.
Now for | y | 1, we have y x , thus
FY ( y ) Pcfw_Y y Pcfw_ X y FX ( y ) .
1
1
1

EL6303 Probability and Stochastic Processes
XK Chen, NYU/ECE
Introduction
Probability and Stochastic processes is an interesting branch of
mathematics that deals with measuring or determining quantitatively the
likelihood that an event occurs. In this cou

EL6303
1. Given
HW 8
X (t )
Ecfw_X (t )
(a) Find
Fall 2016
is a WSS (wide sense stationary) process with
(constant) and
.
RX (t1,t2 ) RX ( ), where t1 t2
(b) Find
. Let
Y (t ) X '(t )
.
.
Ecfw_Y (t )
Ecfw_X (t)Y (t )
(c) Are X(t) and Y(t) uncorrelated? P

Lecture 6
6.3 Two functions of two random variables
Z g ( X ,Y )
W h( X , Y )
FZW ( z, w) Pcfw_Z z,W w
f XY ( x, y )dxdy
DZW
y
Dzw
Dzw
x
g ( x, y) z
(boundaries)
h
(
x
,
y
)
w
1
Example
Y
1
Z X Y , W
and f XY ( x, y )
e
2
X
2
Find FZW ( z, w) and f ZW

EL6303
HW 6 Solutions
Fall 2016
1. X and Y have joint density function
f ( x, y)
XY
(1)
(2)
(3)
(4)
1 xy
A
, | x |1,| y |1; zero,otherwise.
Find A so that the above defined is a valid joint density function.
Are X,Y uncorrelated? Show your answer.
Are X,

EL6303
Sample Midterm Test 1 with Solutions
Prof Ted Rappaport
(This was a real Midterm Test 1 in Fall 2013. Use as reference only.)
Part 1 (Long Questions.)
1.
with
Y g(X )
| x |, for 1 x 1
g ( x)
0, otherwise
and
.
( x 1)
f X ( x) e
u( x 1)
(1) Find a

Lecture 5
Chapter 6: Two Random Variables
Suppose A and B are two events. We know that in order to study A
and B, just knowing P ( A) and P ( B) is not enough. We have to know
how they are related to each other. That is we have to know P ( AB ).
Similarly

EL6303
Solution to HW 5
1. X and Y are independent with
Z X Y
. Find
f Z ( z)
f X ( x) u( x 1) u( x 2), fY ( y) u( y) u( y 1)
f Z ( z) f X ( z) fY ( z)
f X ( ) fY ( z )d
(2) Geometric Method.
Solution: (1) Convolution approach. Let us calculate
Z X Y
.
b

APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.8, SOLUTION
Peter Lakner
1. We alredy know from the class notes that q = 1/2, P [Y5 = 3/2] = .3125, P [Y5 =
(3/2)2 ] = .15625, P [Y5 = (3/2)3 ] = .15625, P [Y5 = (3/2)4 ] = .03125, and P [Y5 = (3/2)5 ]

APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.9, SOLUTION
Peter Lakner
Here is the tree for the payoffs (Yt ):
q=
1+rd
ud
= .545 and 1 q = .455. Use the formula
Zt1 = max Yt1 , EQ
Zt
Ft1
.
1.0382
Below is the tree for the prices of this American

APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.1, SOLUTION
Peter Lakner
(a) Below is the tree for the discounted prices:
We need to satisfy G () 0 for every and G () > 0 for at least one . We get the
following three inequalities, corresponding to ea

STERN SCHOOL OF BUSINESS
NEW YORK UNIVERSITY
COURSE SUPPLEMENT
STAT-GB.2219.89 and STAT-UB.0008.01
APPLIED STOCHASTIC PROCESSES
FOR FINANCIAL MODELS
Professor Peter Lakner
Office: Kaufman Management Center 8-61
Phone: (212) 998-0476
Email: plakner@stern.n

EL6303
Solutions to HW 9
1. Suppose
W (t )
is the Wiener process with
RW (t1,t2 ) min(t1,t2 )
Find
Ecfw_X (t)
and
. Define
RX (t1,t2 )
Ecfw_W (t ) 0
X (t ) W (t 1) W (t)
. Is
X (t )
Fall 2015
and
.
WSS?
(Video is required.)
Solution:
Ecfw_X (t ) Ecfw_W (t

EL6303
HW 7
1. Prove or disprove that, if
Fall 2016
Ecfw_X 2 Ecfw_Y 2 Ecfw_XY ,
2. Prove or disprove that, for any X, Y, and
then X Y
in the MS sense.
0, Pcfw_| X Y | 1 Ecfw_| X Y |2
2
3. Find A such that
4. Show that if
AX
an a
5. The random process
2

Lecture 7
Chapter 7 Sequences of Random Variables
7.1 General Concepts
Definition
The random vector is a vector X [x1 , x 2 ,L , x n ] whose components
xi s are random variables.
Definition
The joint distribution is
F ( X ) F ( x1 , x2 ,L , xn ) P x1 x1 ,

EL6303
HW 3
Fall 2016
1. Let
P( X k ) Bkp k 1, k 1,2,., ; 0 p 1.
(a) Find B so that
represents a probability mass function.
P( X k )
(b) Find
Ecfw_X .
(c) Find
Ecfw_X 2.
(d) Find
.
2
2
Ecfw_X E cfw_X
(e) Let
. Find the conditional probability mass funct

EL6303
HW 3 Solution
1. A fair coin is tossed 10,000 times. Find the probability that the number of
heads is between 4,800 and 5,200.
Solution: We know that
if
,
k2 n
npq 1
k nk
k2 np
k1 np
) G(
)
p q G(
k k1 k
are of the order of
npq
npq
.
k1 np and k