Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 262

Figure A.1: Possible sample functions of discrete-space stochastic processes:
(a) continuous-time; (b) discrete-time.

The probabilistic description of a stochastic process is given by means of
either the joint probability distribution function 1 (PDF - also known as the
cumulative distribution function) of the random variables { X(t i ),i = 1, 2, ··· ,n } ,

F X (x;t) = P { X(t 1 ) ≤ x 1 ,X(t 2 ) ≤ x 2 , ··· ,X(t n ) ≤ x n } (A.1)

or, alternatively, of their joint probability density function (pdf),

f X (x;t) = δF X (x;t) (A.2)

δx

The complete probabilistic description of the process X(t) requires the spec-
ification of either one of these two functions for all values of n, and for all
possible n-tuples (t 1 ,t 2 , ··· ,t n ).
The state space of the process can be either discrete or continuous. In the
former case we have a discrete-space stochastic process, often referred to as
a chain. In the latter case we have a continuous-space process. The state

is usually the set of natural integers IN = { 0, 1, 2, ···} , or space of a chain

a subset of it.
If the index (time) parameter t is continuous, we have a continuous-time pro-
cess. Otherwise the stochastic process is said to be a discrete-time process,
and is often referred to as a stochastic sequence, denoted as { X n ,n = 0, 1, 2, ···} .
In this appendix we are interested in stochastic processes with a discrete
state space, both in continuous and in discrete time. For the sake of sim-
plicity, the results of this appendix only refer to the case of finite state
spaces.

1 Note that throughout the topic we use the notation P{A} to indicate the probability
of event A , and boldface to denote row vectors and matrices.