# semimartingale topology

Let $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}}),\mathbb{P})$ 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

$${\int}_{0}^{t}{\xi}^{n}\mathit{d}{X}^{n}-{\int}_{0}^{t}{\xi}^{n}\mathit{d}X\to 0$$ |

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 $\int \xi \mathit{d}{X}^{n}$ converges to $\int \xi \mathit{d}X$, 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

$${X}_{0}^{n}-{X}_{0}+\int {\xi}^{n}\mathit{d}{X}^{n}-\int {\xi}^{n}\mathit{d}X\stackrel{ucp}{\to}0$$ |

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 $\int \xi \mathit{d}X$). 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

$${\lambda}_{n}{\int}_{0}^{t}{\xi}^{n}\mathit{d}X\to 0$$ |

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

$$\{{\int}_{0}^{t}\xi \mathit{d}X:|\xi |\le 1\text{is simple predictable}\}$$ |

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).

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 |