semimartingale convergence implies ucp convergence
Let be a sequence of cadlag adapted processes converging to in the semimartingale topology. Then, converges ucp to .
To show this, suppose that in the semimartingale topology, and define the stopping times by
in probability as . However, note that whenever for some then and . So
as , proving ucp convergence.
As a minor technical point, note that the result that the hitting times are stopping times requires the filtration to be at least universally complete. However, this condition is not needed. It is easily shown that semimartingale convergence is not affected by passing to the completion (http://planetmath.org/CompleteMeasure) of the filtered probability space or, alternatively, it is enough to define the stopping times in (1) by restricting to finite but suitably dense subsets of and using the right-continuity of the processes.
|Title||semimartingale convergence implies ucp convergence|
|Date of creation||2013-03-22 18:40:44|
|Last modified on||2013-03-22 18:40:44|
|Last modified by||gel (22282)|