CT4 Stochastic Processes

Question 1

What is a Stochastic Process? Give an example

Answer 1

A Stochastic process is a time dependent random phenomenon, modeled as a set of ordered random variables, X(t), where t is the time.

Question 2

What is a Counting Process?

Answer 2

A counting process, X(t) is a continuous/discrete time process with S= {0,1,2,3…} and X(t) is a non-decreasing function of time, i.e. X(s) <= X(t), E s < t.

Question 3

Markov Chain?

Answer 3

A Markov chain is a stochastic process that exhibits the Markov property and has a discrete state and discrete time set.

For example, the three state NCD model.

Question 4

Markov Jump Process?

Answer 4

It is a stochastic process that exhibits the Markov property, it has a discrete time set and continuous time set.

For example, the HSD model.

Question 5

Sample Path

Answer 5

The sample path of a stochastic process is a joint realisation of X(t), for all t E J

Question 6

Describe a mixed type process.

Answer 6

A mixed type process is a stochastic process is one such that the Time Set or State Space is both discrete as well as continuous.

Question 7

Give an example of a Mixed process.

Answer 7

A pension Scheme

The number of contributors towards a pension scheme can exit a population either through:

Retirement at the end of each retirement age.
State space= { 100, 99, 98,…,0}
Time Set = { 1, 2, 3, 4, 5 }

Death at any point in time.
State space ={ 100, 99, 98,…,0}
Time Set = {0, infinity}

Bonus point: A mixed type process can be used when trying to model the bank balance of a Car Sales Dealer.

Question 8

A Stochastic process with a discrete time set and continuous state space is?

Answer 8

A Time Series.

Question 9

A Stochastic process with a continuous time set and continuous state space is?

Answer 9

A Compound Process,

An example is General Insurance.

Question 10

White Noise Process

Answer 10

A set of i.i.d ordered random variables.

{X(t)} t E J can for example follow a Normal Distribution or a Binomial Distribution.

Question 11

White Noise Process

Answer 11

A stochastic process that consists of a set of i.i.d random variables. The random variables can be either discrete or continuous and the time set can be either dicrete of continuous.

{X(t)} t E J can for example follow a Normal Distribution or a Binomial Distribution.

Question 12

(General) Random Walk.

Answer 12

in simple terms, it is the sum of the white noise process.
Y(t=0) = 0
Y(t) = sum X(j), where j=1 to j=t

X(j) , is a white noise process.

Question 13

Symmetric Random Walk

Answer 13

It is a simple random walk, where the probabilities of moving up or down are equal. i.e p = 0.5

Y(t)= Y(t-1) +1 , p= 0.5
Y(t-1) -1 , p= 0.5

Question 14

What are the important properties of stochastic processes

Answer 14

• Stationarity- Strict Stationarity
  Weak Stationarity
  -Markov Property
• Increments and their Independence.

Question 15

Markov Property

Answer 15

The future state of a process can be sufficiently predicted by the present state and the past states do not provide additional information for the prediction of the future.

Question 16

Explain what is meant by the Markov property in the context of a stochastic process and in terms of filtrations.

Answer 16

The filtration notation F is used for a continuous time process since it is difficult to list all the values at all past times.

E.g. For a continuous-time process with continuous state space,
P( X(n) E A | F(s)) = P( X(n) E A | X(s) )

Question 17

What is a compound Poisson process

Answer 17

A compound process is
X(t) = sum Y(j) , where j=1 to j=N(t),
the randon variables { Y(j); j=1,2,3…} are i.i.d and usually continuous.