Editing Stochastic process (section)

===Construction issues===
When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes.<ref name="KloedenPlaten2013page63"/><ref name="Khoshnevisan2006page153"/> One problem is that it is possible to have more than one stochastic process with the same finite-dimensional distributions. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions.<ref name="Billingsley2008page493to494">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=493–494}}</ref> This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.<ref name="Adler2010page13"/><ref name="Borovkov2013page529">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|pages=529–530}}</ref>

Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined.<ref name="Ito2006page32"/> For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable.<ref name="AdlerTaylor2009page7"/><ref name="Khoshnevisan2006page153"/> For a continuous-time stochastic process <math>X</math>, other characteristics that depend on an uncountable number of points of the index set <math>T</math> include:<ref name="Ito2006page32"/>
* a sample function of a stochastic process <math>X</math> is a [[continuous function]] of <math>t\in T</math>;
* a sample function of a stochastic process <math>X</math> is a [[bounded function]] of <math>t\in T</math>; and
* a sample function of a stochastic process <math>X</math> is an [[increasing function]] of <math>t\in T</math>.
where the symbol '''∈''' can be read "a member of the set", as in <math>t</math> a member of the set <math>T</math>.

To overcome the two difficulties described above, i.e., "more than one..." and "functionals of...", different assumptions and approaches are possible.<ref name="Asmussen2003page408"/>