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T. Krak
composite system as its components wear out, break down and get replaced; or the
spread of pathogens through a population; or the evolution of stock prices—and so
on and so forth.
What all these systems have in common is that there is a dynamic component to
their description—they change over time—and they are, in a sense, hard to describe
exactly. For instance, this difficulty may arise because their behaviour depends on
unknown external influences or because the system cannot reasonably be described
at a sufficiently detailed level. Thus, there arises an uncertainty about how exactly
the system will evolve over time, even if one can model how it will ‘roughly’ behave.
Regardless of the interpretation that we want to assign to this uncertainty, such
systems are modelled using stochastic processes. A stochastic process, then, is a
probabilistic description of the system under study. In this sense, it provides a formal
and integrated description of the system dynamics and the probabilistic uncertainty
of its evolution.
On the other hand, we might also be uncertain about whether such a model
is ‘correct’. For instance, we might not know exactly the numerical values that
the parameters of our model should take. Similarly, we might be aware that our
modelling assumptions lead to simplifications that are not necessarily warranted,
which introduces uncertainty about the accuracy or applicability of any assessments
made on the basis of these models. It is therefore of interest to robustify our models
also against these kinds of ‘meta’, or ‘higher-order’, uncertainties.
In this chapter, we consider stochastic processes for which this higher-order
uncertainty is modelled using the theory of imprecise probabilities (IP). For an
extended introduction to IP, we refer the reader back to Chap. 2. We constrain
ourselves to briefly recalling that such imprecise probabilistic models can be
interpreted as representing a set of traditional probabilistic models. So, in our
current setting, we will be considering sets of stochastic processes. From an
inference point of view, the aim is then to compute inferences which are robust
with respect to variations within such a set. We recall from Chap. 2 that these robust
inferences are captured in general by the lower and upper expectations with respect
to the elements of the set that we are considering.
Our aim with the present chapter is to provide an extensive but intuitive
introduction to the theory of imprecise stochastic processes and of imprecise
Markov chains in particular. To this end, we will intentionally focus on the different
representations of these processes. We will show how each of the different ways
of looking at these models provides its own way of deriving useful properties and
highlights different intuitive ways of reasoning about them. Important results and
properties are stated, but we have made an effort to keep the discussion intuitive.
We try to prevent technicalities and do not provide extended proofs; instead, we will
provide pointers to the literature that the interested reader might pursue herself.
The remainder of this chapter is organised as follows. We start the discussion
by giving a quick introduction to stochastic processes in Sect. 5.2. The first part
basically uses the measure-theoretic approach (albeit in a rather simplified sense)
to pin down some first concepts and notation. We then go on to present three
different and graphical representations of stochastic processes, which can be used
T. Krak
composite system as its components wear out, break down and get replaced; or the
spread of pathogens through a population; or the evolution of stock prices—and so
on and so forth.
What all these systems have in common is that there is a dynamic component to
their description—they change over time—and they are, in a sense, hard to describe
exactly. For instance, this difficulty may arise because their behaviour depends on
unknown external influences or because the system cannot reasonably be described
at a sufficiently detailed level. Thus, there arises an uncertainty about how exactly
the system will evolve over time, even if one can model how it will ‘roughly’ behave.
Regardless of the interpretation that we want to assign to this uncertainty, such
systems are modelled using stochastic processes. A stochastic process, then, is a
probabilistic description of the system under study. In this sense, it provides a formal
and integrated description of the system dynamics and the probabilistic uncertainty
of its evolution.
On the other hand, we might also be uncertain about whether such a model
is ‘correct’. For instance, we might not know exactly the numerical values that
the parameters of our model should take. Similarly, we might be aware that our
modelling assumptions lead to simplifications that are not necessarily warranted,
which introduces uncertainty about the accuracy or applicability of any assessments
made on the basis of these models. It is therefore of interest to robustify our models
also against these kinds of ‘meta’, or ‘higher-order’, uncertainties.
In this chapter, we consider stochastic processes for which this higher-order
uncertainty is modelled using the theory of imprecise probabilities (IP). For an
extended introduction to IP, we refer the reader back to Chap. 2. We constrain
ourselves to briefly recalling that such imprecise probabilistic models can be
interpreted as representing a set of traditional probabilistic models. So, in our
current setting, we will be considering sets of stochastic processes. From an
inference point of view, the aim is then to compute inferences which are robust
with respect to variations within such a set. We recall from Chap. 2 that these robust
inferences are captured in general by the lower and upper expectations with respect
to the elements of the set that we are considering.
Our aim with the present chapter is to provide an extensive but intuitive
introduction to the theory of imprecise stochastic processes and of imprecise
Markov chains in particular. To this end, we will intentionally focus on the different
representations of these processes. We will show how each of the different ways
of looking at these models provides its own way of deriving useful properties and
highlights different intuitive ways of reasoning about them. Important results and
properties are stated, but we have made an effort to keep the discussion intuitive.
We try to prevent technicalities and do not provide extended proofs; instead, we will
provide pointers to the literature that the interested reader might pursue herself.
The remainder of this chapter is organised as follows. We start the discussion
by giving a quick introduction to stochastic processes in Sect. 5.2. The first part
basically uses the measure-theoretic approach (albeit in a rather simplified sense)
to pin down some first concepts and notation. We then go on to present three
different and graphical representations of stochastic processes, which can be used