semimartingale topology

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}),\mathbb{P})$ be a filtered probability space and $(X^{n}_{t})$, $(X_{t})$ be cadlag adapted processes. Then, $X^{n}$ is said to converge to $X$ in the semimartingale topology if $X^{n}_{0}\rightarrow X_{0}$ in probability and

 $\int_{0}^{t}\xi^{n}\,dX^{n}-\int_{0}^{t}\xi^{n}\,dX\rightarrow 0$

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

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.

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

 $X^{n}_{0}-X_{0}+\int\xi^{n}\,dX^{n}-\int\xi^{n}\,dX\xrightarrow{\rm ucp}0$

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

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

 $D^{\rm ucp}(X)=\sum_{n=1}^{\infty}2^{-n}\mathbb{E}\left[\min\left(1,\sup_{t

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

 $D^{\rm s}(X)=\sup\left\{D^{\rm ucp}(X_{0}+\xi\cdot X):|\xi|\leq 1\textrm{ is % simple previsible}\right\}$

($\xi\cdot X$ 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 $\lambda_{n}\not=0$ is a sequence of real numbers converging to zero and $X$ is a cadlag adapted process then $\lambda_{n}X\rightarrow 0$ in the semimartingale topology if and only if

 $\lambda_{n}\int_{0}^{t}\xi^{n}\,dX\rightarrow 0$

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

 $\left\{\int_{0}^{t}\xi\,dX:|\xi|\leq 1\textrm{ is simple predictable}\right\}$

is bounded    in probability for every $t>0$. So, $\lambda_{n}X\rightarrow 0$ 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).

Title semimartingale topology SemimartingaleTopology 2013-03-22 18:40:41 2013-03-22 18:40:41 gel (22282) gel (22282) 6 gel (22282) Definition msc 60G48 msc 60G07 msc 60H05 semimartingale convergence UcpConvergence UcpConvergenceOfProcesses