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T. Krak
on (Ω, F ). Given this probability space, we can finally formalise the notion of a
stochastic process as a collection {X t } t∈T of random variables associated to this
probability space. We will here slightly restrict our definition to the following
specific stochastic process:
Definition 5.1 (Stochastic process) Fix a time-dimension T and consider a probability space (Ω, F , P ). Then (the corresponding) stochastic process is the collection {X t } t∈T of random variables X t : Ω → X : ω → ω(t), t ∈ T, on this
space.
Corollary 5.1 Fix a time-dimension T; consider a probability space (Ω, F , P );
and let {X t } t∈T be the corresponding stochastic process. Then for all t ∈ T and
x ∈ X , it holds that Pr(X t = x) = P
{ω ∈ Ω : ω(t) = x}
.
Proof Fix t ∈ T, and recall the definition of a random variable: for all x ∈ X , the
probability Pr(X t = x) of X t taking the value x is equal to P
X
−1
t (x)
, the measure
of its preimage in Ω. Since X t (ω) = ω(t), we have X
−1
t (x) = {ω ∈ Ω : ω(t) =
x}.
The above is a formal way of saying that, and how, these random variables {X t } t∈T
are associated to the given probability space. In words, for some fixed time t ∈ T,
X t is a random variable that takes on a value x ∈ X with probability equal to the
measure of the set of paths along which the state at time t is x. Conversely, if we fix
the outcome ω ∈ Ω, then the collection {X t } t∈T can be considered a deterministic
process, and X t (ω) = ω(t) for all t ∈ X .
Note, therefore, that all the quantitative information about the probability of the
process taking on certain values at given points in time are completely determined
by the measure P . It is therefore also intuitive to instead consider this measure P to
be ‘the stochastic process’, although this is technically an abuse of terminology. This
is because, for a given probability space (Ω, F , P ), it is possible to define many
different stochastic processes; any T-indexed collection of random variables on this
space satisfies the general definition. However, in a sense, the stochastic process in
Definition 5.1 can be viewed as the ‘canonical’ stochastic process corresponding
to the given probability space, since it specifically and exactly represents the
uncertainty about which states might be obtained at different points in time. We
will therefore, and for notational convenience, often refer to the measure P and its
corresponding stochastic process {X t } t∈T interchangeably and without confusion.
Next, it will be convenient to have a standardised notation to index a subset of the
random variables of a stochastic process. To this end, for any finite sequence of time
points t = t 1 , . . . , t n in T, with n ∈ N, we will write X t = X t 1 , . . . , X t n . Typically,
these sequences will be taken to be ordered, so that t 1 < · · · < t n . Note that each of
the random variables X t i , i = 1, . . . , n takes values in X . Hence, the sequence X t
takes values (jointly) in X n = × n
i=1 X . An element of this joint state-space is thus
a vector (x 1 , . . . , x n ) ∈ X n . When we are explicitly talking about a sequence t of
n time points, we will also write x t to denote a generic element of X n .
T. Krak
on (Ω, F ). Given this probability space, we can finally formalise the notion of a
stochastic process as a collection {X t } t∈T of random variables associated to this
probability space. We will here slightly restrict our definition to the following
specific stochastic process:
Definition 5.1 (Stochastic process) Fix a time-dimension T and consider a probability space (Ω, F , P ). Then (the corresponding) stochastic process is the collection {X t } t∈T of random variables X t : Ω → X : ω → ω(t), t ∈ T, on this
space.
Corollary 5.1 Fix a time-dimension T; consider a probability space (Ω, F , P );
and let {X t } t∈T be the corresponding stochastic process. Then for all t ∈ T and
x ∈ X , it holds that Pr(X t = x) = P
{ω ∈ Ω : ω(t) = x}
.
Proof Fix t ∈ T, and recall the definition of a random variable: for all x ∈ X , the
probability Pr(X t = x) of X t taking the value x is equal to P
X
−1
t (x)
, the measure
of its preimage in Ω. Since X t (ω) = ω(t), we have X
−1
t (x) = {ω ∈ Ω : ω(t) =
x}.
The above is a formal way of saying that, and how, these random variables {X t } t∈T
are associated to the given probability space. In words, for some fixed time t ∈ T,
X t is a random variable that takes on a value x ∈ X with probability equal to the
measure of the set of paths along which the state at time t is x. Conversely, if we fix
the outcome ω ∈ Ω, then the collection {X t } t∈T can be considered a deterministic
process, and X t (ω) = ω(t) for all t ∈ X .
Note, therefore, that all the quantitative information about the probability of the
process taking on certain values at given points in time are completely determined
by the measure P . It is therefore also intuitive to instead consider this measure P to
be ‘the stochastic process’, although this is technically an abuse of terminology. This
is because, for a given probability space (Ω, F , P ), it is possible to define many
different stochastic processes; any T-indexed collection of random variables on this
space satisfies the general definition. However, in a sense, the stochastic process in
Definition 5.1 can be viewed as the ‘canonical’ stochastic process corresponding
to the given probability space, since it specifically and exactly represents the
uncertainty about which states might be obtained at different points in time. We
will therefore, and for notational convenience, often refer to the measure P and its
corresponding stochastic process {X t } t∈T interchangeably and without confusion.
Next, it will be convenient to have a standardised notation to index a subset of the
random variables of a stochastic process. To this end, for any finite sequence of time
points t = t 1 , . . . , t n in T, with n ∈ N, we will write X t = X t 1 , . . . , X t n . Typically,
these sequences will be taken to be ordered, so that t 1 < · · · < t n . Note that each of
the random variables X t i , i = 1, . . . , n takes values in X . Hence, the sequence X t
takes values (jointly) in X n = × n
i=1 X . An element of this joint state-space is thus
a vector (x 1 , . . . , x n ) ∈ X n . When we are explicitly talking about a sequence t of
n time points, we will also write x t to denote a generic element of X n .