Problem Set 6: Answer key
1) The model is:
yt + xt = u1t
u1t = 0:2u1t
yt + xt = u2t
u2t = u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. The order the integration can be found by noting that u1t is I(1):
1
0:2z
P (z)
=
1
0:2z
0:8z 2
P (1)
=
1
0:2
0:8 = 0
0:
Problem set 0: Prior notions
+1
Let fyt gt=
1
be a sequence of scalar bounded random variables.
1)
Let L and 1 be the "lag" and "identity" operators respectively,
dened as follows:
Lyt
Lk yt
L0 yt
L0
=
=
=
=
yt 1
L(Lk
yt
1
1
)yt = yt
k
Show that L is a li
Problem Set 8: Answer key
1) We shall begin with computing conditional expectations.
E (st =st
1
= 1)
E (st =st
1
= 0)
= P (st = 1=st
= p
= P (st = 1=st
= 1 q
1
= 1)
1
= 0)
We can combine both in the following formula:
E (st =st
1)
p
=
1
st
st
q
1
1
=1
=0
Problem Set 7: Answer key
1) Our model is:
yt
"t
=
+ "t
= ut ( 0 +
t=+1
1
fut gt=
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
We shall begin by computing the conditional moments, as the unconditional ones can then be obtained by way of the Law of Iterated Expectation
Problem Set 1: Answer Key
1) To obtain the process for yt is very simple:
"t
= yt
=
(yt
=
(1
yt
yt
1
) + !t
) + yt + ! t
+1
So, fyt gt= 1 is also a stationary AR(1) process with the same autore+1
gressive coe cient as f"t gt= 1 . However, as the "mutatis
Problem Set 4: Answer key
1) In exercise 3 of Problem Set 3 we had the following process:
yt =
Let
2
0:02
0:03
(L) = I
0:5 0:1
0:4 0:5
+
yt
2
2L ,
1L
1
+
0
0:25
0
0
yt
2
+ ut
where:
1
=
0:5
0:4
0:1
0:5
2
=
0
0:25
0
0
Consider the VMA(1) representation (wh
Problem Set 2: Box-Jenkins methodology
1) For an AR(1) process we have:
(0)
=
2
"
=
(0)
2
"
1
2
1
2
Hence,
2
p lim R2 =
For a MA(1) process,
(0)
2
"
(0)
=
2
(1 +
1
1+
=
)
2
"
2
Therefore,
p lim R2
=
1
1+
1
2
2
=
1+
2
We can see that a consistent estimator
Problem Set 3: Answer Key
1) Lets write the univariate AR(p) process as:
yt = c +
1 yt 1
+ : +
p yt p
+ "t
Like we did in exercise 6 of Problem Set 0, we shall build a system with
the equation that describes the evolution of yt and the following p 1 ident
Problem Set 7: ARCH processes
1) For the following ARCH(1) model:
yt
"t
t=+1
1
fut gt=
=
+ "t
= ut ( 0 +
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
compute the rst through fourth moments for yt both conditional on yt
unconditional.
1
and
2) For the following model:
Problem Set 5: Answer key
1) The claim is not true. A process that is stationary in the strict sense and
that has nite second moments is covariance stationary. But a process
that is strictly stationary can be non covariance stationary. Lets construct a pr
Problem Set 9: Answer Key
A state representation consists of a state equation and an observation
equation. To obtain it, we must bare in mind that the state variable, t , has to
be an unobservable random variable. In addition, letting vt and ! t represent
Problem Set 4: VAR Analysis
1) For the bivariate VAR(2) process in exercise 3 of Problem Set 3 compute the
impulse responses s for s= 1,2 as well as the matrix long run multiplier.
2) Provide an example to show that Granger causality is not transitive.
3)
Problem Set 8: Markov Chains
For the following Markov Chain process:
+1
fst gt= 1 :
p (st = 1=st 1 = 1) =
p (st = 0=st 1 = 0) =
st 2 S = f0; 1g 8t
p
q
1) Compute the conditional and unconditional expectation and variance of st .
2) Compute the expected nu
Problem Set 1: ARMA processes
1) Consider the following three stochastic processes:
yt
=
+ "t
"t
=
"t
1 + !t
+1
f! t gt= 1
is a White Noise
+1
Show that the autoregressive behaviour of f"t gt= 1 is passed (mutatis
+1
+1
mutandis) on to fyt gt= 1 ; i.e. fy
Problem Set 6: Cointegration
1) Consider the following example:
yt + xt
= u1t
yt + xt
= u2t
where:
u1t
=
0:2u1t
u2t
=
u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. What is the order of integration of yt and xt ?
ii. Under which conditions are yt and xt coi
Problem Set 3: Introduction to VAR processes
1) Show that the companion form of a scalar AR(p) process is a p-variate
VAR(1) process
2) For a stationary n-variate VAR(p) process:
i. show that its companion form is an np-variate VAR(1) process.
3) Consider
Problem Set 9: Kalman lter
1) Provide the state-space representation for the following (stationary) arma
processes:
a. yt =
+ ut + u t
1
b. yt = c + yt
1
+ ut
c. yt = c + yt
1
+ ut + u t
1
2) For the series rf in mpeppext.w and the series cons in weldat.w
Problem Set 2: Box-Jenkins methodology
1) Recall the denition of R2 :
T
X
t=1
T
X
R2 = 1
(yt
b
"2t
y)2
t=1
+1
For a consistent estimator of the parameters of the process fyt gt=
p lim R2 = 1
1
:
2
"
0
+1
Compute the last expression when fyt gt= 1 follows
Problem Set 5: Unit Root Processes
1) Prove the following claim or nd a counterexample. "Every Strict Stationary
process is Covariance Stationary"
t=+1
1:
2) Consider the following two alternative processes for fyt gt=
A)
B)
t=+1
1
where f"t gt=
yt
yt
= a
Chapter 10
Hilbert spaces
This chapter contains some non-examinable, though useful, foundational material on Hilbert spaces. Hilbert spaces play an important rle in the rigorous
o
construction of the Ito integral in the next chapter. Some basic knowledge
Chapter 12
Conditional expectations
and martingales
12.1
Conditional expectations
To start with, we take another look at conditional probabilities. You might want
to consult Section 2.3 to refresh your memory: there we introduced conditional
probability d
Chapter 8
Measure-theoretic
probability
In this chapter we will try to familiarize you with the language of measuretheoretic probability theory. A probability will be seen as a map giving dierent
weights, between 0 and 1, to possible future events or outc
MATHEMATICAL METHODS
AUTUMN 2009
LECTURE 9: THE ITO STOCHASTIC INTEGRAL
(LECTURE NOTES, CHAPTERS 11 (AND 10)
1. Introduction
We will assume from now that our Brownian motion (Wt )t0 is dened
on some probability space (, F, P) (for which one may take BM if
Mathematical Methods for
Financial Engineering, I
Autumn 2009
Raymond Brummelhuis
Department of Economics, Mathematics and Statistics,
Birkbeck College, University of London,
Malet Street, London WC1E 7HX
October 1, 2009
2
Chapter 1
Introduction
Market pr
MATHEMATICAL METHODS
AUTUMN 2009
LECTURE 8: MEASURE THEORETIC PROBABILITY
(SUMMARY OF LECTURE NOTES, CHAPTERS 8 AND 9)
RAYMOND BRUMMELHUIS
BIRKBECK
Probability Spaces (lecture Notes, sections 8.1 - 8.3)
Measure theoretic probability: axiomatics based on t
Chapter 6
Multivariable Ito Calculus
As is the case for ordinary Calculus, there exists a multi-variable version of
Ito Calculus involving more than one Brownian. It is relevant for modelling
situations in which there are several independent sources of un
Chapter 3
Brownian motion
3.1
Introducing Brownian motion
Brownian motion has become one of the fundamental building blocks of modern
quantitative nance. Indeed, the basic continuous-time model for nancial asset
prices assumes that the log-returns of a gi
Chapter 4
A crash course in It
o
calculus
In this chapter we introduce the basic rules of stochastic calculus or It calculus.
o
We will do this using an intuitive approach which is based on calculus-style
dierentials, postponing the mathematically rigorou
Chapter 7
Characteristic functions of
random variables
We briey leave the subject of stochastic processes to get acquainted with a
powerful tool for analysing sums of independent random variables, the characteristic function of a random variable. To some
Chapter 5
Stochastic processes and
PDEs
There is a close connection between solving an sde and solving boundary value
problems for a certain type of partial dierential equations (pdes) which are
known as parabolic. Parabolic pdes are, roughly speaking, th