Introduction to Imprecise Probabilities

Chapter 11Stochastic processes

Filip Hermans¹ and Damjan Škulj²

¹SYSTeMS Research Group, Ghent University, Belgium

²Faculty of Social Sciences, University of Ljubljana, Slovenia

11.1 The classical characterization of stochastic processes

11.1.1 Basic definitions

In many applications we are interested in the evolution of an uncertain value. For example, we may want to study the evolution of a stock on the stock market over a certain period or we may be interested in the inter arrival time between packages on a router or it might be interesting to know how the radio-active decay of an isotope evolves over time. This however means that we have to incorporate dynamics in our modelling.

The evolution will be addressed using an index. Often this index is the time, but this is not necessarily so. Think for example about a sequence of nucleotides in a chromosome which can be considered as a random sequence. Throughout the majority of this chapter we will assume the index to be discrete. This does not necessarily mean that the actual dynamics behind the studied uncertain value have to be of discrete nature as well. The discrete index is usually more like an observation index than the representation of the real dynamics behind the process which is just sampled at regular times.

In the rest of this section we give the basic definitions of random processes with the emphasis on Markov chains. In the next section the so-called event driven approach to random processes is presented, which ...