Semimartingale

As semimartingales certain processes are referred to in the stochastics that are particular to the definition of a general stochastic integral of importance. The class of semimartingales includes many well-known stochastic processes, such as the Wiener process ( Brownian motion ) or the Poisson process.

• 3.1 Martingale
• 3.2 jump processes
• 3.3 Ito processes

Definition

Given a probability space with associated filtration. It is assumed that the filtration is complete ( all P - zero levels are measurable).

A semimartingale is then a stochastic process with values ​​in with:

• Is adapted to,
• The paths / trajectories of càdlàg are so right-continuous and left-sided Limites exist,
• There exists a (not necessarily unique) representation: which almost certainly a local martingale and a process of locally finite variation is finite and measurable.

Properties

Stochastic integration

As already indicated in the introduction, can be constructed general stochastic integrals with the help of semimartingales. Semimartingales represent the largest class of integrators, for an integral of the form

Can be meaningfully defined. in this case comes from the set of all locally bounded predictable processes.

Stability under transformations

The class of semimartingales is stable under many operations. Not only is each stopped semimartingale obviously again a semimartingale, also under localization, a " change of time" or a transition to a new absolutely continuous measure remain semimartingales.

Examples

Martingale

Every martingale is trivially a semimartingale, since every martingale is a local martingale itself.

In addition, each submartingale a semimartingale and any supermartingale if it is right-continuous with left-sided limits exist.

Jump processes

Many jump processes as generalized Poisson processes are semimartingales, as they are of bounded variation.

Ito processes

Among other things, in financial mathematics Ito processes play a central role. These are represented as

Where the last term an Ito integral denoted by the volatility process. This term is a local martingale.