Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

2Revision of probability and stochastic processes

2.1 Revision of probabilistic concepts

Consider a probability space , where is a measurable space and is a probability defined on it. Usually, it is a model for a real‐world phenomenon or an experiment that depends on chance (i.e. is random) and we shall now see what each element of the triplet means.

The universal set or sample space is a non‐empty set containing all possible conditions that may influence the outcome of the random phenomenon or experiment.

If we throw two dice simultaneously, say one white and one black, and are interested in the outcome (number of dots on each of the two dice), the space could be the set of all possible ‘physical scenarios’ describing the throwing of the dice, such as the position of the hands, how strongly and in what direction we throw the dice, the density of the air, and many other factors, some ...