Stochastic processes can be seen as random variables taking their values in a function space. It is therefore important to understand the naturallly associated -algebras.
Let , , be the set of functions . We denote by the -algebra generated by the so-called cylindrical sets
where and where are products of intervals: .
As a -algebra is also generated by the following families:
- where and where are Borel sets in .
- where and where is a Borel set in .
Exercise: Show that the following sets are not in :
The above exercise shows that the -algebra is not rich enough to include natural events; this is due to the fact that the space is by far too big.
In these lectures, we shall mainly be interested in processes with continuous paths. In that case, we use the space of continuous functions endowed with the -algebra that is generated by
and where are products of intervals . This -algebra enjoys nice properties. It is for instance generated by the open sets of the (metric) topology of uniform convergence on compact sets.
The -algebra is generated by the open sets of the topology of uniform convergence on compact sets.
We make the proof when the dimension and let the reader adapt it in higher dimension. Let us first recall that, on the topology of uniform convergence on compact sets is given by the distance
This distance endows with the structure of a complete, separable, metric space (that is of a Polish space). Let us denote by the -field generated by the open sets of this metric space.
First, it is clear that the cylinders
are open sets that generate . Thus, we have
On the other hand, since for every , , and ,
we deduce that
Since is generated by the above sets, this implies
and concludes the proof.
Show that the following sets are in :