# completeness of semimartingale convergence

###### Theorem.

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$ be a filtered probability space. Then, the space of semimartingales $\mathcal{S}$ forms a complete (http://planetmath.org/Complete) topological vector space under the semimartingale topology.

That is, semimartingale convergence is a vector topology (http://planetmath.org/TopologicalVectorSpace) and, for any sequence $X^{n}\in\mathcal{S}$ such that $X^{n}-X^{m}\rightarrow 0$ as $m,n\rightarrow\infty$ then there exists an $X\in\mathcal{S}$ with $X^{n}\rightarrow X$.

Title completeness of semimartingale convergence CompletenessOfSemimartingaleConvergence 2013-03-22 18:40:47 2013-03-22 18:40:47 gel (22282) gel (22282) 4 gel (22282) Theorem msc 60G07 msc 60G48 msc 60H05