proof of completeness under ucp convergence
Let be a filtered probability space, be a sub--algebra of , and be a set of real valued functions on which is closed (http://planetmath.org/Closed) under uniform convergence on compacts. We show that both the set of -measurable processes and the set of jointly measurable processes with sample paths almost surely in are complete (http://planetmath.org/Complete) under ucp convergence. The method used will be to show that we can pass to a subsequence which almost surely converges uniformly on compacts.
We start by writing out the metric generating the topology of uniform convergence on compacts (compact-open topology) for functions . This is the same as uniform convergence on each of the bounded intervals for positive integers ,
Then, the metric is . Convergence under the ucp topology is given by
for any jointly measurable stochastic process , with the (pseudo)metric being .
Now, suppose that is a sequence of jointly measurable processes such that as . Then, and we may pass to a subsequence satisfying whenever . So,
In particular, this shows that is almost surely finite and, therefore,
as , with probability one.
So, the sequence is almost surely Cauchy (http://planetmath.org/CauchySequence), under the topology of uniform convergence on compacts. We set
As measurability of real valued functions is preserved under pointwise convergence, it follows that if are -measurable, then so is . In particular, is a jointly measurable process. Furthermore, since convergence is almost surely uniform on compacts, if have sample paths in with probability one then so does .
It only remains to show that . However, we have already shown that with probability one, hence .
|Title||proof of completeness under ucp convergence|
|Date of creation||2013-03-22 18:40:35|
|Last modified on||2013-03-22 18:40:35|
|Last modified by||gel (22282)|