In this chapter we give fundamental definitions of probabilistic concepts. Since the theory is more transparent in the discrete case, it is presented first. The most important concepts not met in elementary courses are the models for information, its flow and conditional expectation. This is only a brief description, and a more detailed treatment can be found in many books on Probability Theory, for example, Breiman (1968), Loeve (1978), Durret (1991). Conditional expectation with its properties is central for further development, but some of the material in this chapter may be treated as an appendix.

2.1 Discrete Probability Model

A probability model consists of a filtered probability space on which variables of interest are defined. In this section we introduce a discrete probability model by using an example of discrete trading in stock.

Filtered Probability Space

A filtered probability space consists of: a sample space of elementary events, a field of events, a probability defined on that field, and a filtration of increasing subfields.

Sample Space

Consider a single stock with price S _t at time t = 1, 2, T. Denote by ? the set of all possible values of stock during these times.

If we assume that the stock price can go up by a factor u and down by a factor d, then the relevant information reduces to the knowledge of the movements at each time.

To model uncertainty about the price in the future, we "list" all possible future...