University of Pretoria

WST312 Stochastic Processes

Past Questions Memo

Chapter 1 and 2
Question 1 a) For a stochastic process { X t : t ∈ [0, ∞)} define the Markov property.

b) What is a stochastic process having the Markov property with a discrete parameter space
called? Markov chain
c) What is a stochastic process having the Markov property with a continuous parameter space
called? Markov Jump process

Question 2 Consider a branching process with X 0 = 1 i.e. the zero-th generation has only
one element. Let µ be the expected number of offspring of any element and let σ 2 be the
variance of the number of offspring of any element. Let X n be the size of the n th generation.
a) Show that E[ X n ] = µ n .

b) Give a formula for the variance of X n in terms of µ , σ 2 and the variance of X n −1 .

c) What can you say about the expected size of the n th generation as n → ∞ if µ < 1 ?
Question 3 What does it mean if a stochastic process { X t : t ∈ T } has the following
properties? In each case also state one of the following processes which possesses the relevant
property:

White noise, Random walk, Moving average process, Poisson process, Compound
Poisson process, Brownian motion

a) Stationarity

c) Weak Stationarity

d) Independent increments

Question 4 You are thinking of moving to Australia and as part of your investigation of life
on the other side of the ocean you have collected the following sets of data:
1) the maximum daily temperature each day since 1 January 2003
2) whether or not it rained for each day since 1 January 2003
3) the number of cyclists injured in road accidents since 1 January 2003

For each data set choose an appropriate state space and parameter space, indicating for each if
it is discrete or continuous.

1) Parameter Space: each day since 1 Jan 2003, discrete i.e. t = 1,2,3,...
State Space: maximum daily temperature, ℜ : continuous (technically) or (−∞, ∞) : discrete
(more realistic)
2) Parameter Space: each day since 1 Jan 2003, discrete i.e. t = 1,2,3,...
State Space: yes or no i.e. two states, discrete
3) Parameter Space: every possible time since 1 Jan 2003, continuous
State Space: cumulative number of cyclists injured, discrete i.e. S = {0,1,2,...} (counting
process)

Question 5 Consider a moving average process of order 1.

(a) Prove that this process is weakly stationary.
1
Yt = ∑ α i X t −i = α 0 X t + α 1 X t −1 , for t = 0,1,2,... and where { X t : t = −1,0,1,2,3,...} is white
i =0

noise and α 0 , α 1 are real numbers.

WEAKLY STATIONARY: all Yt ’s have the same expected values and is covariance
stationary(i.e. cov(Yt , Ys ) depends only on t − s ).

Let E [X t ] = µ since { X t : t = −1,0,1,2,3,...} is white noise.

Then E [Yt ] = E [α 0 X t + α 1 X t −1 ] = α 0 µ + α 1 µ which is independent of t therefore every
Yt has the same expected value. (1 MARK)

Now let var( X t ) = σ 2 , then for s ≤ t ,

cov(Yt , Ys ) = cov(α 0 X t + α 1 X t −1 , α 0 X s + α 1 X s −1 )
= α 0 cov( X t , X s ) + α 0α 1 cov( X t , X s −1 ) + α 0α 1 cov( X t −1 , X s ) + α 1 cov( X t −1 , X s −1 )
2 2

0 if t > s + 1
 2 2
= α 0 σ + α 1 σ 2 if t = s
2


α 0α 1σ if s + 1 = t
2

(1/2 MARK)
which depends only an how far t and s are and not the actual times. (1/2 MARK)

(b) Does this process have the Markov property? Prove your answer.
For any t , Yt = α 0 X t + α 1 X t −1 , and Yt −1 = α 0 X t −1 + α 1 X t − 2 , thus since Yt −1 depends on
X t − 2 , which appears in Yt − 2 , we have that Yt is dependent on at least Yt − 2 and therefore
the Markov Property does not hold. (2 MARK)

(c) Does this process have independent increments? Prove your answer.
Yt +1 − Yt = α 0 X t +1 + α 1 X t − α 0 X t − α 1 X t −1 = α 0 X t +1 + (α 1 − α 0 ) X t − α 1 X t −1 and
Yt −1 = α 0 X t −1 + α 1 X t − 2 thus since the increment depends on X t −1 and Yt −1 depends on X t −1 ,
Yt +1 − Yt does depend on {Ys : 0 ≤ s ≤ t} and we do not have independent increments. (2
MARKS)
Question 6 Consider a branching process with X 0 = 1 i.e. the zeroth generation has only one
element. Let µ be the expected number of offspring of any element and let σ 2 be the variance
of the number of offspring of any element. Let X n be the size of the n th generation. Prove that
σ 2 µ n −1 (1 − µ n )
 if µ ≠ 1
E[ X n ] = µ n and var( X n ) =  1− µ .
nσ 2 if µ = 1

Proof

Question 7 a) Define fully a stochastic process, also describing all the elements that make it
up.

b) Is the following statement true or false? Explain your answer.

To specify the probability structure of a stochastic process {X t : t ∈ T } it is required to
specify the distribution of X t for all values of t ∈ T .

c) Choose from the following list,

Random Walk, Moving Average process, Poisson process, Compound Poisson process,
Branching process
one process that satisfies each of the following properties. Show that the property is indeed
satisfied.
(i) Stationarity

(ii) Independent Increments

 (iii) Markov Property

d) For a compound Poisson process with parameter λ , prove that the moment generating
function is given by e λt ( M X (u ) −1) .
Proof

Question 8 a) Explain what is meant by a stationary stochastic process.

A stochastic process {X t : t ∈ T } is a stationary process if for any integer n , any values
t1 < t 2 < ... < t n which are all elements of T and any value k such that k + t1 , k + t 2 ,..., k + t n
are all elements of T , it is true that the joint distribution of X t1 , X t2 ,..., X tn is the same as
the joint distribution of X t1 + k , X t2 + k ,..., X tn + k .

b) Give an example of a process which is not stationary. Show that the process given is
indeed not stationary.
The random walk is not stationary:
 t 
If we let E [X t ] = µ ≠ 0 then E[Yt ] = E ∑ X i  = tµ which depends on t . Thus if the
 i =1 
expected values are not equal the distributions cannot be equal either.

Question 9 a) Define what it means for a stochastic process {X t : t ∈ T } to have independent

increments.
the increment in the value of the process over any interval t to t + s (i.e. X t + s − X t ) is
independent of what happened before or at time t i.e. the increment is independent of
{X u : 0 ≤ u ≤ t}.
b) What does it mean for a process {X t : t ∈ T } to have stationary increments? Give an example
of a process which has stationary increments.
This means that the distribution of the increments X t + s − X t depends only on the distance
between the two random variables i.e. s time units.
e − λ (t − s ) (λ (t − s ) )
n
The Poisson process: P[N t − N s = n] = for n = 0,1,2,...
n!
Brownian motion: Each increment Bt − Bs , s < t , has a normal distribution with expected
value µ (t − s ) and variance σ 2 (t − s ) , where µ and σ 2 > 0 are constants.

Question 10 A commodity is being stocked to satisfy a continuous demand. The inventory

policy is described by specifying two non-negative critical values s and S , with s < S . The
inventory level is checked periodically at fixed times t 0 , t1 , t 2 ,... . If the stock on hand at t n is
less than or equal to s , then by immediate procurement the stock level is brought up to level
S . On the other hand if the stock level is greater than s , then no replenishment is done. Let
Z n be the total demand for the commodity during the interval [t n −1 , t n ) , and let X n be the stock
on hand just before time t n .
a) Complete the following in terms of S, X n and Z n :
 __________________________________ if s < X n ≤ S , Z n +1 ≤ X n

X n +1 =  __________________________________ if X n ≤ s, Z n +1 ≤ S
 __________________________________
 otherwise
 X n − Z n +1 if s < X n ≤ S , Z n +1 ≤ X n

X n +1 = S − Z n +1 if X n ≤ s, Z n +1 ≤ S
0
 otherwise
b) Is the parameter space for { X n } discrete or continuous? Explain.
Discrete, since we define X n be the stock on hand just before time t n and t 0 , t1 , t 2 ,... are
period fixed times.

c) What is the state space for the process { X n } ?

{0,1,2,..., S }

d) Define the Markov property for a general process {Yn } .

If for any a ∈ S , for any value of n and for any values of t 0 < t1 < ... < t n < t n +1 , it is true
that

P[Yn +1 = a | Y0 = y 0 , Y1 = y1 ,..., Yn = y n ] = P[Yn +1 = a | Yn = y n ] .

e) In terms of Z n , when will { X n } have the Markov property?

When P[ Z n +1 = k | X 0 ,..., X n ] = P[ Z n +1 = k | X n ] i.e. if the demand in [t n , t n +1 ) only depends
on the stock just before time t n and not times t 0 , t1 ,..., t n −1 .

Question 11 a) Fully define what it means for a stochastic process { X t : t ∈ T } to have

independent increments.
b) Fill in the blank with a ⇒, ⇐ or ⇔ .

Markov property ________________________ Independent increments

t
c) Consider the random walk process {Yt : t = 0,1,2,...} where Y0 = 0 and Yt = ∑ X i for all
i =1

t ≥ 1 with {X t : t = 0,1,2,...} a white noise process. Show that the random walk possesses the
Markov property.

Question 12 For each of the following processes, provide the state space and parameter space
and state whether they are each discrete, continuous or either. Explain as necessary.
Question 13 Define the terms stationarity, weak stationarity and covariance stationarity.
Also give an example of process(es) which possess these properties and explain why it has
the relevant property.

Question 14 Indicate whether the statements that follow are true or false. If false, explain
why, or correct the statement.
a) To specify the probability structure of a stochastic process {X t : t ∈ T } with the X t ’s
independent, it is required to specify the distribution of each X t for all values of t ∈ T .
True

b) A stochastic process {X t : t ∈ T } is stationary if joint distribution of X t1 , X t2 ,..., X tn is the

same as the joint distribution of X t1 +1 , X t2 +1 ,..., X tn +1 for any integer n , and any values
t1 < t 2 < ... < t n which are all elements of T .
False, it must hold for any integer n , any values t1 < t 2 < ... < t n which are all elements
of T and any value k such that k + t1 , k + t 2 ,..., k + t n are all elements of T , it is true that
 the joint distribution of X t1 , X t2 ,..., X tn is the same as the joint distribution of
X t1 + k , X t2 + k ,..., X tn + k .

c) A covariance stationary process is also weakly stationary.

False, the process has to have all the X t ’s with the same expected value for it to be
weakly stationary if it is already covariance stationary.

d) For a process {X t : t ∈ T }, if X t + s − X t is independent of {X u : 0 ≤ u ≤ t}, then the process

has stationary increments.
False, the process has independent increments

Question 15 Suppose that { X n : n = 0,1,2,3,...} is a branching process with X 0 = 1 . Let Φ(z )

be the probability generating function of the number of offspring of any single element, S i .
∞
Let Φ n ( z ) = ∑ P[ X n = i ]z i . Prove that Φ n +1 ( z ) = Φ n (Φ ( z )) .
i =0
Proof

Question 16 a) Fully define a stochastic process, clearly indicating the spaces involved.

b) Consider a mathematical model f (t ) = 2e −9t for t ∈ [0.1,1.1] , which is aimed at modelling

the probability of not contracting a deadly virus if exposed for t minutes. Explain the difference
between a stochastic and deterministic model using the given model, including how it could be
adjusted to be of the other kind. Your explanation should make clear the difference between a
stochastic and deterministic model.

Question 17 a) For a stochastic process { X t : t ∈ T } , define the following properties, clearly

indicating their differences.

Stationarity

Independent Increments

Stationary Increments
 Time-homogeneity

b) For a Poisson process with parameter λ describe which of the properties in (a) hold and
explain why.

c) Derive the moment generating function for a compound Poisson process with parameter λ
. Proof

Question 19 a) Let {Yn : n = 0,1, 2...} be a Markov chain with state space S ⊆ {0,1,2,...}.
i. Show that E (Yn | Y1 , Y2 ,..., Ym ) = E (Yn | Ym ) .
 ii. State the two properties that must hold in order to classify this process as a
martingale.

b) Let the processes { X n } and {Yn } be defined as follows:

n
P( X n =1) =P( X n =−1) =0.5 for n = 0,1, 2,... and Yn = ∑ X i , with Y0 = 0 .
i =0

i. What are the processes { X n } and {Yn } called?

ii. Show that { X n } is not a martingale, while {Yn } is a martingale.

Question 20 Consider a moving average process of order 2.

a) Show that this process is weakly stationary.
b) What can be said of the variance of a weakly stationary process? Justify your answer.
c) To specify the probability structure of such a stochastic process, the distribution of each
Yt , t ∈ T is required. TRUE or FALSE? Explain.

Question 21 a) Define a Poisson process with parameter λ .

b) Provide formulas for the expected value and variance of a compound Poisson process,
including any conditions necessary.

Question 22 a) Consider the following two functions where t ∈ [0,1] :

a if t < 0.4 a with probability t
f (t ) =  , g (t ) =  .
b if t ≥ 0.4 b with probability 1 − t

Using these two functions, explain the concept of a deterministic and a stochastic model
respectively.
b) Consider a stochastic process { X t : t ∈ T } . Define and describe the parameter and state
space.
Question 23 Consider a stochastic process { X t : t ∈ T } .
a) Define the property of stationarity.
b) Define the property of covariance stationarity.
c) Prove that a stationary stochastic process is also covariance stationary.
d) Show that a Poisson process with parameter λ is not covariance stationary. Hint: For
t > s, var( N t − N s ) = cov( N t − N s , N t − N s ) = var( N t ) + var( N s ) − 2 cov( N t , N s ) .