ABSTRACT
In this chapter we shall consider processes continuous in time, and with continuous state spaces. In previous chapters we have discussed processes where both stage and state are discrete: random walks (Chapter 3) where the stage is the number of steps and the state is the position of the walk and where the stage is continuous and the state is discrete: for example, birth processes (Chapter 6) where the stage is time and the state is the number of births. Note that unlike birth processes there is no possibility of an “event” or movement not occurring in any time interval. The processes developed here may be defined in terms of a continuous random variable X(t) which depends on continuous time t > 0 $ t>0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math10_1.tif"/> , although extension to more than a one-dimensional random variable is possible, and as shown below has been useful in modelling certain physical behaviours. However, we shall be explaining the one-dimensional case in detail.
