in probability as , for every and sequence of simple predictable processes .
This topology occurs with stochastic calculus where, according to the dominated convergence theorem (http://planetmath.org/DominatedConvergenceForStochasticIntegration), stochastic integrals converge in the semimartingale topology. Furthermore, stochastic integration with respect to any locally bounded (http://planetmath.org/LocalPropertiesOfProcesses) predictable process is continuous under the semimartingale topology. That is, if are semimartingales converging to then converges to , a fact which does not hold under weaker topologies such as ucp convergence.
Also, for cadlag martingales, convergence implies semimartingale convergence.
It can be shown that semimartingale convergence implies ucp convergence. Consequently, converges to in the semimartingale topology if and only if
for all sequences of simple predictable processes .
The topology is described by a metric as follows. First, let be a metric defining the ucp topology. For example,
Then, a metric for semimartingale convergence is given by
If is a sequence of real numbers converging to zero and is a cadlag adapted process then in the semimartingale topology if and only if
in probability, for every and simple predictable processes . By the sequential characterization of boundedness (http://planetmath.org/SequentialCharacterizationOfBoundedness), this is equivalent to the statement that
is bounded in probability for every . So, in the semimartingale topology if and only if is a semimartingale. It follows that semimartingale convergence only becomes a vector topology (http://planetmath.org/TopologicalVectorSpace) when restricted to the space of semimartingales. Then, it can be shown that the set of semimartingales is a complete topological vector space (http://planetmath.org/CompletenessOfSemimartingaleConvergence).
|Date of creation||2013-03-22 18:40:41|
|Last modified on||2013-03-22 18:40:41|
|Last modified by||gel (22282)|