A sequence of stochastic processes, , is said to converge to a process X under the semimartingale topology, as n goes to infinity, if the following conditions are met. First, should tend to in probability. Also, for every sequence of elementary predictable processes with ,
in probability for all times t. For short, this will be denoted by .
The semimartingale topology is particularly well suited to the class of semimartingales, and to stochastic integration. Previously, it was shown that the cadlag and adapted processes are complete under semimartingale convergence. In this post, it will be shown that the set of semimartingales is also complete. That is, if a sequence of semimartingales converge to a limit X under the semimartingale topology, then X is also a semimartingale.
Theorem 1 The space of semimartingales is complete under the semimartingale topology.
The same is true of the space of stochastic integrals defined with respect to any given semimartingale. In fact, for a semimartingale X, the set of all processes which can be expressed as a stochastic integral can be characterized as follows; it is precisely the closure, under the semimartingale topology, of the set of elementary integrals of X. This result was originally due to Memin, using a rather different proof to the one given here. The method used in this post only relies on the elementary properties of stochastic integrals, such as the dominated convergence theorem.
Theorem 2 Let X be a semimartingale. Then, a process Y is of the form for some if and only if there is a sequence of bounded elementary processes with .
Writing S for the set of processes of the form for bounded elementary , and for its closure under the semimartingale topology, the statement of the theorem is equivalent to