Semimartingale Completeness

A sequence of stochastic processes, {X^n}, is said to converge to a process X under the semimartingale topology, as n goes to infinity, if the following conditions are met. First, {X^n_0} should tend to {X_0} in probability. Also, for every sequence {\xi^n} of elementary predictable processes with {\vert\xi^n\vert\le 1},

\displaystyle  \int_0^t\xi^n\,dX^n-\int_0^t\xi^n\,dX\rightarrow 0

in probability for all times t. For short, this will be denoted by {X^n\xrightarrow{\rm sm}X}.

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 {X^n} 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 {\int\xi\,dX} 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 {Y=\int\xi\,dX} for some {\xi\in L^1(X)} if and only if there is a sequence {\xi^n} of bounded elementary processes with {\int\xi^n\,dX\xrightarrow{\rm sm}Y}.

Writing S for the set of processes of the form {\int\xi\,dX} for bounded elementary {\xi}, and {\bar S} for its closure under the semimartingale topology, the statement of the theorem is equivalent to

\displaystyle  \bar S=\left\{\int\xi\,dX\colon \xi\in L^1(X)\right\}. (1)

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U.C.P. and Semimartingale Convergence

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 {f_n\colon{\mathbb R}_+\rightarrow{\mathbb R}} converges uniformly on compacts to a limit {f} if it converges uniformly on each bounded interval {[0,t]}. That is,

\displaystyle  \sup_{s\le t}\vert f_n(s)-f(s)\vert\rightarrow 0 (1)

as {n\rightarrow\infty}.

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 {X^n} converges to the limit {X} uniformly on compacts in probability if

\displaystyle  {\mathbb P}\left(\sup_{s\le t}\vert X^n_s-X_s\vert>K\right)\rightarrow 0

as {n\rightarrow\infty} for each {t,K>0}.

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