semimartingale topology

Let (Ω,,(t)t+),) be a filtered probability space and (Xtn), (Xt) be cadlag adapted processes. Then, Xn is said to converge to X in the semimartingale topology if X0nX0 in probability and


in probability as n, for every t>0 and sequence of simple predictable processes |ξn|1.

This topologyMathworldPlanetmathPlanetmath occurs with stochastic calculus where, according to the dominated convergence theorem (, stochastic integralsMathworldPlanetmath converge in the semimartingale topology. Furthermore, stochastic integration with respect to any locally bounded ( predictable process ξ is continuous under the semimartingale topology. That is, if Xn are semimartingales converging to X then ξ𝑑Xn converges to ξ𝑑X, a fact which does not hold under weaker topologies such as ucp convergence.

Also, for cadlag martingalesMathworldPlanetmath, L1 convergence implies semimartingale convergence.

It can be shown that semimartingale convergence implies ucp convergence. Consequently, Xn converges to X in the semimartingale topology if and only if


for all sequences of simple predictable processes |ξn|1.

The topology is described by a metric as follows. First, let Ducp(X-Y) be a metric defining the ucp topology. For example,


Then, a metric Ds(X-Y) for semimartingale convergence is given by

Ds(X)=sup{Ducp(X0+ξX):|ξ|1 is simple previsible}

(ξX denotes the integral ξ𝑑X). This is a proper metric under identification of processes with almost surely equivalentPlanetmathPlanetmath sample paths, otherwise it is a pseudometric.

If λn0 is a sequence of real numbers converging to zero and X is a cadlag adapted process then λnX0 in the semimartingale topology if and only if


in probability, for every t>0 and simple predictable processes |ξn|1. By the sequential characterization of boundedness (, this is equivalent to the statement that

{0tξ𝑑X:|ξ|1 is simple predictable}

is boundedPlanetmathPlanetmathPlanetmathPlanetmath in probability for every t>0. So, λnX0 in the semimartingale topology if and only if X is a semimartingale. It follows that semimartingale convergence only becomes a vector topology ( when restricted to the space of semimartingales. Then, it can be shown that the set of semimartingales is a complete topological vector space (

Title semimartingale topology
Canonical name SemimartingaleTopology
Date of creation 2013-03-22 18:40:41
Last modified on 2013-03-22 18:40:41
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Definition
Classification msc 60G48
Classification msc 60G07
Classification msc 60H05
Synonym semimartingale convergence
Related topic UcpConvergence
Related topic UcpConvergenceOfProcesses