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T. Krak
whence the parametrisation now requires an entire family Q s of rate matrices—
one for each point in time. Note, though, that these matrices are still transition
rate matrices, in that they satisfy the properties in Definition 5.16. However, the
corresponding matrix differential equation is no longer solved by a simple matrix
exponential.
More generally still, for arbitrary continuous-time stochastic processes (that are
neither homogeneous nor Markov) we may consider the transition rates (derivatives)
not only for specific points in time but also for specific histories leading up to that
time. For instance, with s = s 1 , . . . , s n and t in R ≥0 and x s ∈ X n , we may write
d
d u
P
X u = y | X s = x s , X t = x
u=t
=: Q x s ,t (x, y) .
(5.8)
Thus, the parametrisation requires the specification of a transition rate matrix for
each point in time and for each possible history before that time. It should be clear
that this leads to a rather unwieldy process specification, which again goes some
way in illustrating why homogeneity and Markovianity are such popular simplifying
assumptions.
5.4.1 Imprecise Continuous-Time Markov Chains
With the notation and concepts for precise continuous-time stochastic processes in
place, let us now turn to the imprecise generalisation. In what follows, we will
consider imprecise, homogeneous continuous-time Markov chains (ICTMC). As
before, we start by considering the abstract sets-of-measures definition:
Definition 5.17 (ICTMC as set of processes) An imprecise continuous-time
Markov chain is a set P of probability measures on the measurable space (Ω, F ) of
(continuous-time) paths, with associated lower expectation operator E P such that,
for all f ∈ L (X ) and all s 1 , . . . , s n , t ∈ R ≥0 such that s 1 < · · · < s n < t, it holds
that
E P
f (X t )
X s 1 , . . . , X s n
= E P
f (X t )
X s n
.
Furthermore, an imprecise continuous-time Markov chain is called homogeneous
if, for all s, t ∈ R ≥0 , s < t, and all f ∈ L (X ), it holds that E P
f (X t )
X s
=
E P
f (X t−s )
X 0
.
As in the discussion about imprecise discrete-time Markov chains, we distinguish
between the definition by epistemic irrelevance—which is what is used above—and
the definition by strong independence, which would imply that all P ∈ P are precise
(homogeneous) Markov chains, and which we are explicitly not using.
Let us now consider the parametrisation of such an ICTMC. We recall that in
the precise case, the canonical parameter is a single transition rate matrix Q. In
T. Krak
whence the parametrisation now requires an entire family Q s of rate matrices—
one for each point in time. Note, though, that these matrices are still transition
rate matrices, in that they satisfy the properties in Definition 5.16. However, the
corresponding matrix differential equation is no longer solved by a simple matrix
exponential.
More generally still, for arbitrary continuous-time stochastic processes (that are
neither homogeneous nor Markov) we may consider the transition rates (derivatives)
not only for specific points in time but also for specific histories leading up to that
time. For instance, with s = s 1 , . . . , s n and t in R ≥0 and x s ∈ X n , we may write
d
d u
P
X u = y | X s = x s , X t = x
u=t
=: Q x s ,t (x, y) .
(5.8)
Thus, the parametrisation requires the specification of a transition rate matrix for
each point in time and for each possible history before that time. It should be clear
that this leads to a rather unwieldy process specification, which again goes some
way in illustrating why homogeneity and Markovianity are such popular simplifying
assumptions.
5.4.1 Imprecise Continuous-Time Markov Chains
With the notation and concepts for precise continuous-time stochastic processes in
place, let us now turn to the imprecise generalisation. In what follows, we will
consider imprecise, homogeneous continuous-time Markov chains (ICTMC). As
before, we start by considering the abstract sets-of-measures definition:
Definition 5.17 (ICTMC as set of processes) An imprecise continuous-time
Markov chain is a set P of probability measures on the measurable space (Ω, F ) of
(continuous-time) paths, with associated lower expectation operator E P such that,
for all f ∈ L (X ) and all s 1 , . . . , s n , t ∈ R ≥0 such that s 1 < · · · < s n < t, it holds
that
E P
f (X t )
X s 1 , . . . , X s n
= E P
f (X t )
X s n
.
Furthermore, an imprecise continuous-time Markov chain is called homogeneous
if, for all s, t ∈ R ≥0 , s < t, and all f ∈ L (X ), it holds that E P
f (X t )
X s
=
E P
f (X t−s )
X 0
.
As in the discussion about imprecise discrete-time Markov chains, we distinguish
between the definition by epistemic irrelevance—which is what is used above—and
the definition by strong independence, which would imply that all P ∈ P are precise
(homogeneous) Markov chains, and which we are explicitly not using.
Let us now consider the parametrisation of such an ICTMC. We recall that in
the precise case, the canonical parameter is a single transition rate matrix Q. In