proof of completeness of semimartingale convergence

We start by showing that semimartingale convergence is a vector topology ( on the space of semimartingales. That is, addition of processes and multiplication by real numbers are continuous ( operationsMathworldPlanetmath.

The continuity of addition follows immediately from the fact that the topology is given by a translation-invariant metric Ds; if XnX and YnY then


so Xn+YnX+Y. Also, suppose that λn are real numbers converging to λ. Then, it is easily shown that Ds(λnY)max(|λn|,1)Ds(Y) for all processes Y and, therefore,


As was noted (in semimartingale convergence (, the statement that D((λn-λ)X)0 whenever λnλ is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the statement that X is a semimartingale, so λnXnλX and Ds does indeed generate a vector topology.

It only remains to show that the topology is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. So, suppose that Xn-Xm0 under the semimartingale topology. Then, we also have ucp convergence (see semimartingale convergence implies ucp convergence) and, XnucpX for a cadlag adapted process X (see completeness under ucp convergence). We need to show that this also converges under the semimartingale topology.

For any simple predictable process ξ, 0tξ𝑑Xn0tξ𝑑X in probability. Therefore, for any sequence of simple predictable processes ξn,


where the limit is taken in probability. If |ξn|1 then the condition that Xn-Xm goes to zero in the semimartingale topology implies that the right hand side tends to zero in probability as n goes to infinityMathworldPlanetmath and, XnX in the semimartingale topology.

This shows that semimartingale convergence is complete on the space of cadlag adapted processes. To show that it is complete on the set of semimartingales, we just need to show that the process X above is a semimartingale whenever Xn are. However, for any sequence of real numbers λn0 then,


As noted previously, the condition that Xm are semimartingales gives λnXm0 as n tends to infinity.

lim supnDs(λnX)supnmax(|λn|,1)Ds(X-Xm).

Taking the limit m shows that λnX tends to zero in the semimartingale topology and, consequently, X is a semimartingale.

Title proof of completeness of semimartingale convergence
Canonical name ProofOfCompletenessOfSemimartingaleConvergence
Date of creation 2013-03-22 18:40:54
Last modified on 2013-03-22 18:40:54
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 60G07
Classification msc 60G48
Classification msc 60H05