Semimartingale
In stochastics , semimartingales are certain processes that are particularly important for the definition of a general stochastic integral . The semimartingale class includes many well-known stochastic processes such as the Wiener process ( Brownian motion ) or the Poisson process .
definition
A probability space with an associated filtration is given . It is assumed that the filtration is complete (all P zero quantities can be measured).
A semimartingale is then a stochastic process with values in with:
- is adapted to
- the paths / trajectories of are càdlàg , i.e. continuous on the right and the limits on the left exist,
- there is a (not necessarily unambiguous) representation: where it is almost certainly finite and measurable, a local martingale and a process of locally finite variation .
properties
Stochastic integration
As already indicated in the introduction, general stochastic integrals can be constructed with the help of semimartingals . Semimartingals represent the largest class of integrators for which an integral of the form
can be meaningfully defined. in this case comes from the set of all locally restricted, predictable processes .
Stability under transformations
The semimartingale class is stable under many operations. Not only is every stopped semimartingale obviously a semimartingale again, even under localization, a "change of time" or a transition to a new absolutely constant measure , semimartingales are preserved.
Examples
Martingales
Every martingale is trivially a semimartingale, since every martingale is itself a local martingale.
In addition, every submartingale is a semimartingale and every supermartingale , provided it is continuous to the right with limit values existing on the left .
Jump processes
Many jump processes , such as generalized Poisson processes, are semimartingals because they are of limited variation.
Ito processes
Ito processes play a central role in financial mathematics , among other things . These can be represented as
where the last term denotes an Ito integral with volatility process. This term is a local martingale.
literature
- Jean Jacod, Albert N. Shiryaev: Limit Theorems for Stochastic Processes . 2nd Edition. Springer, Berlin 2002, ISBN 3-540-43932-3 .
- Philip Protter: Stochastic Integration and Differential Equations . 2nd Edition. Springer, Berlin 2003, ISBN 3-540-00313-4 .