semimartingale topology

Let $(\mathrm{\Omega },ℱ,{({ℱ}_t)}_{t\in {ℝ}_{+}}),\mathbb{P})$ be a filtered probability space and $(X_t^n)$, $(X_t)$ be cadlag adapted processes.Then, $X^n$ is said to converge to $X$ in the semimartingale topology if $X_0^n\to X_0$ in probability and

$${\int }_0^t{\xi }^n𝑑X^n-{\int }_0^t{\xi }^n𝑑X\to 0$$

in probability as $n\to \mathrm{\infty }$, for every $t>0$ and sequence of simple predictable processes $|{\xi }^n|\le 1$.

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 $\xi$ is continuous under the semimartingale topology. That is, if $X^n$ are semimartingales converging to $X$ then $\int \xi 𝑑X^n$ converges to $\int \xi 𝑑X$, a fact which does not hold under weaker topologies such as ucp convergence.

Also, for cadlag martingales, $L^1$ convergence implies semimartingale convergence.

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

$$X_0^n-X_0+\int {\xi }^n𝑑X^n-\int {\xi }^n𝑑X\stackrel{ucp}{\to }0$$

for all sequences of simple predictable processes $|{\xi }^n|\le 1$.

The topology is described by a metric as follows. First, let $D^{\mathrm{ucp}}(X-Y)$ be a metric defining the ucp topology. For example,

Then, a metric $D^{\mathrm{s}}(X-Y)$ for semimartingale convergence is given by

$$D^{\mathrm{s}}(X)=sup\{D^{\mathrm{ucp}}(X_0+\xi \cdot X):|\xi |\le 1\text{ is simple previsible}\}$$

($\xi \cdot X$ denotes the integral $\int \xi 𝑑X$). This is a proper metric under identification of processes with almost surely equivalent sample paths, otherwise it is a pseudometric.

If ${\lambda }_n\ne 0$ is a sequence of real numbers converging to zero and $X$ is a cadlag adapted process then ${\lambda }_{n}X\to 0$ in the semimartingale topology if and only if

$${\lambda }_n{\int }_0^t{\xi }^n𝑑X\to 0$$

in probability, for every $t>0$ and simple predictable processes $|{\xi }^n|\le 1$.By the sequential characterization of boundedness (http://planetmath.org/SequentialCharacterizationOfBoundedness), this is equivalent to the statement that

$$\{{\int }_0^t\xi 𝑑X:|\xi |\le 1\text{ is simple predictable}\}$$

is bounded in probability for every $t>0$.So, ${\lambda }_{n}X\to 0$ in the semimartingale topology if and only if $X$ 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).