, where can be (deterministic), (random, independent of ), or (random, dependent on ).3 3 This construction is also described in Muldowney [115].
4 4 Recall also that I1 and I2 make reference to a construction .
5 5 See [MTRV] for discussion of complex‐valued random variables.
6 6 There are other important stochastic integrals, such as .
Chapter 2 Random Variation
2.1 What is Random Variation?
The previous chapter makes reference to random variables as functions which are measurable with respect to some probability domain. This conception of random variation is quite technical, and the aim of this chapter is to illuminate it by focussing on some fundamental features.
In broad practical terms, random variation is present when unpredictable outcomes can, in advance of actual occurrence, be estimated to within some margin of error. For instance, if a coin is tossed we can usually predict that heads is an outcome which is no more or no less likely than tails. So if an experiment consists of ten throws of the coin, it is no surprise if the coin falls heads‐up on, let us say, between four and six occasions. This is an estimated outcome of the experiment, with estimated margin of error.
In fact, with a little knowledge of binomial probability distributions, we can predict that there is approximately 40 per cent chance that heads will be thrown on four, five or six occasions out of the ten throws. So if a ten‐throw trial is repeated one hundred times, the outcome should be four, five, or six heads for approximately four hundred of the one thousand throws.
Such knowledge enables us to estimate good betting odds for placing a wager that a toss of the coin will produce this outcome. This is the “naive or realistic” view.
Can this fairly easily understandable scenario be expressed in the technical language of probability theory, as in Chapter 1 above? What is the probability space
The following remarks are intended to provide a link between the “naive or realistic” view, and the “sophisticated or mathematical” interpretation of this underlying reality.
The possible outcomes of an experiment consisting of a single throw of the coin are H (for heads) and T (for tails). Suppose a sample space
and define a probability measure
Then, trivially,
The set of outcomes of a single throw of a coin is the set
and
Then, for
There are many different ways of defining the probability space. It is natural to use real‐number‐valued functions, so the outcomes
