Lecture 1 - Stochastic Processes

Question 1

What is a model?

Answer 1

A model is an imitation of a real world system or process. Models of many
activities can be developed

Question 2

A model is an imitation of a real world system or process. Models of many
activities can be developed, for example:

Answer 2

◦ the economy,
◦ an insurance company solvency,
◦ a pension scheme deficit.

Question 3

Key steps when developing and using a model include the following:

Answer 3

1. Develop a well-defined set of objectives that need to be met by the modelling process.
2. Plan the modelling process and how the model will be validated. Validation will include
   some tests to ensure the objectives are being met.
3. Collect and analyse the necessary data for the model including assigning appropriate
   values to parameters not determined by the data.
4. Define the model by capturing the essence of the real world system, involving experts on the
   real world system you are modelling.
5. Decide on the simulation package to use when modelling the real world. Write the computer
   program for the model and debug to ensure it is working as expected.
6. Test the reasonableness of the outputs from the model.
7. Review the appropriateness of the model in the light of small changes to the input
   parameters – sensitivity testing.
8. Analyse output and communicate the results to the client.

Question 4

Modelling – the benefits and limitations

What are the advantages of models?

Answer 4

• The main benefit of modelling is that systems with a long time frame (e.g.
  pension schemes) can be studied in a compressed timeframe.
• Complex systems can be modelled using stochastic models which include an
  element of randomness.
• Different future strategies can be modelled and the projected outcomes
  compared to see which output best suits the user.
• The variance of the results can be reduced by controlling the experimental
  conditions when modelling a complex system.

Question 5

Modelling – the benefits and limitations

What are the disadvantages of models?

Answer 5

• Model development requires a significant investment of time and resources.
• Stochastic models will only give estimates of model outputs. Several runs will
  need to be carried out in order to test the outputs appropriately.
• Models can create a false sense of confidence. Even if results look impressive
  the model itself may not reflect reality as accurately as it could.
• Models depend heavily on the data inputs.
• Important to understand the limitations of the model and to be able to
  interpret the results appropriately.

Question 6

What is a stochastic process?

Answer 6

A stochastic process is a collection of random variables indexed by time

Question 7

What must an accurate stochastic model imitate?

Answer 7

The random nature of the variables

Question 8

A model that does not have a random element to it is called a…

Answer 8

deterministic model

Question 9

What is a deterministic model?

Answer 9

Deterministic models are fixed once the input parameters, and relationships
between parameters, have been established. The output is a single answer
with no random element to it.

Question 10

What are the state spaces?

Answer 10

The sets from which the time variable and the value that the process takes can be taken from.

Question 11

What are the time space and state space?

Answer 11

The state space is the space of values that the model can take at a particular
point in time.
The time space is the set of times at which the model is observed.