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). ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ ${\displaystyle ({\mathcal {F}}_{t})}$ ${\displaystyle {\mathcal {F}}_{0}}$

A semimartingale is then a stochastic process with values ​​in with: ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{d}}$

• ${\displaystyle X}$is adapted to ${\displaystyle ({\mathcal {F}}_{t})}$
• the paths / trajectories of are càdlàg , i.e. continuous on the right and the limits on the left exist, ${\displaystyle X}$
• 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 .
  ${\displaystyle X=X_{0}+M+A\,,\;}$
  ${\displaystyle X_{0}}$ ${\displaystyle {\mathcal {F}}_{0}}$ ${\displaystyle M}$ ${\displaystyle A}$

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

${\displaystyle (H\cdot X)_{t}:=\int _{0}^{t}H_{s}dX_{s}}$

can be meaningfully defined. in this case comes from the set of all locally restricted, predictable processes . ${\displaystyle H}$

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

${\displaystyle X_{t}=X_{0}+\int _{0}^{t}b_{s}\,{\rm {d}}s+\int _{0}^{t}\sigma _{s}{\rm {d}}W_{s},}$

where the last term denotes an Ito integral with volatility process. This term is a local martingale. ${\displaystyle \sigma _{s}}$

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 .