The subjects discussed in this appendix are all primarily of a mathematical nature, and constitute background material for the main text. The topics are included for the readers easy reference.

Probability distribution

The notation for probability theory is discussed, and in particular the definition of conditional probability. The concept of a martingale is discussed as this has important applications in finance.

Random variables are real- or discrete-valued variables that take values in some prespecified range determined by their probability distributions. Random variables are designated by either upper case such as X or lower case such as r. A stochastic process refers to a collection of random variables. The stochastic process can be (a) a continuous process with an independent random variable r(t) for every t, with the continuous label t in some range t ∈ [t0, t*], or can be (b) a discrete process with a collection of random variables Zn with n: integer. Degrees of freedom, in the terminology of physics, refer to the number of independent random variables at a given instant, and hence each degree of freedom corresponds to a independent stochastic process.