A mode of convergence on the space of processes which occurs often in the study of stochastic calculus, is that of uniform convergence on compacts in probability or ucp convergence for short.
First, a sequence of (non-random) functions converges uniformly on compacts to a limit if it converges uniformly on each bounded interval . That is,
If stochastic processes are used rather than deterministic functions, then convergence in probability can be used to arrive at the following definition.
Definition 1 A sequence of jointly measurable stochastic processes converges to the limit uniformly on compacts in probability if
as for each .
The notation is sometimes used, and is said to converge ucp to . Note that this definition does not make sense for arbitrary stochastic processes, as the supremum is over the uncountable index set and need not be measurable. However, for right or left continuous processes, the supremum can be restricted to the countable set of rational times, which will be measurable. In fact, for jointly measurable processes, it can be shown that the supremum is measurable with respect to the completion of the probability space, so ucp convergence makes sense. However, that is not needed for these notes.
For each time , the following pseudometric can be defined on the space of locally bounded deterministic processes
Then, uniformly on compacts if for each . This shows that uniform convergence on compacts is in fact given by the following metric
Similarly, a sequence of stochastic processes converges ucp to if in probability. By bounded convergence, this is equivalent to tending to zero. So, ucp convergence is given by the following metric.
If a sequence of processes converges ucp, then it is always possible to pass to a subsequence for which uniform convergence on compacts holds with probability one on the sample paths. This is a simple application of the Borel-Cantelli lemma, but allows properties of ucp convergence to be inferred from corresponding properties of uniform convergence on compacts.
Theorem 2 The space of cadlag (resp. continuous) adapted processes is complete under ucp convergence.
Furthermore, if then there is a subsequence whose sample paths almost surely converge to those of uniformly on compacts.
Proof: Let be a Cauchy sequence under ucp convergence, so that as . Then, there is a subsequence satisfying for all . In this case,
so that is almost surely finite. Restricting to a set of probability one if necessary, we suppose that this is always finite. Then, the sample paths of are Cauchy convergent under uniform convergence on compacts, and there is a limit . As , this will be measurable, so is a stochastic process and is adapted whenever are adapted. If the processes have left limits, then it is clear that and, therefore, the left limits are also Cauchy under uniform convergence on compacts.
As limits can be commuted with uniform convergence of sequences, if the processes are cadlag then,
So, is cadlag and under uniform convergence on compacts. In particular, is continuous whenever are.
To complete the proof, it just remains to show that the original sequence does indeed converge ucp to . As it follows that this converges in probability, so . Then,
as required. ⬜
A consequence of Doob’s martingale inequalities is that convergence of martingales implies ucp convergence.
Lemma 3 Let be a sequence of cadlag martingales, and be a process such that as .
Then, is a martingale and has a cadlag version which is the ucp limit of .
Proof: Clearly, by -convergence, is a martingale. Furthermore, Doob’s martingale inequality shows that
as for all . Consequently, the sequence is Cauchy under ucp convergence and has a cadlag limit . As (convergence in probability), this limit is a cadlag version of . ⬜
In particular, the space of continuous martingales is complete under convergence.
Corollary 4 Let be a sequence of continuous martingales such that as for all and some process . Then, is a martingale and has a continuous version.
Proof: By the previous lemma, is a martingale and has a cadlag version such that . Then, by Theorem 2, this limit is continuous. ⬜
The semimartingale topology
An even stronger topology than ucp convergence is the semimartingale topology. A sequence of cadlag and adapted processes converges to under this topology if, for every sequence of elementary predictable processes with and every , the limit
holds, in probability. Then, a sequence of cadlag adapted processes converges to a limit under the semimartingale topology if converges to zero. As with ucp convergence, this can be described by a metric. For each set
Then, if . It follows that the semimartingale topology is defined by the metric
The semimartingale topology is indeed stronger than ucp convergence.
Theorem 5 If are cadlag adapted processes with converging to in the semimartingale topology then .
Proof: If is a cadlag adapted process and , consider the stopping times
for each positive integer . On the event that , by right-continuity we have for sufficiently large . Hence,
for any real . However, the processes are elementary and . So,
It follows that if in the semimartingale topology then
as , and ucp convergence holds. ⬜
Lemma 3 above states that convergence of martingales implies ucp convergence. In fact, the stronger property of semimartingale convergence holds, although I do not give a proof here (the reader should be able to prove this fact using the martingale inequality established in the later post on martingales as integrators).
Completeness of ucp convergence can be used to also prove completeness of the semimartingale topology.
Lemma 6 The space of cadlag and adapted processes is complete under the semimartingale topology.
Proof: Let be Cauchy under the semimartingale topology. By Theorem 5 this is also Cauchy under ucp convergence, so has a limit which is cadlag and adapted.
As in probability for each time , it follows that in probability, for all elementary processes . So, setting ,
Taking the supremum over all such elementary processes gives
which, by Cauchy convergence, goes to zero as . ⬜
Finally, the semimartingale topology is not a vector topology on the space of cadlag adapted processes. For that to be the case, we would need for all such processes and sequences of real numbers . However, this says that in probability for all elementary processes . Equivalently, for each , the set
is bounded in probability. The processes for which this holds are precisely the semimartingales, although I do not prove that here. On these processes, semimartingale convergence is indeed a vector topology.