Stochastic process
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. 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. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.

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. For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable. For a continuous-time stochastic process X {\displaystyle X}, other characteristics that depend on an uncountable number of points of the index set T {\displaystyle T} include:

where the symbol ∈ can be read "a member of the set", as in t {\displaystyle t} a member of the set T {\displaystyle T}.

To overcome the two difficulties described above, i.e., "more than one..." and "functionals of...", different assumptions and approaches are possible.
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