semimartingale
Semimartingales are adapted stochastic processes which can be used as integrators in the general theory of stochastic integration. Examples of semimartingales include Brownian motion
, all local martingales
, finite variation processes and Levy processes
.
Given a filtered probability space , we consider real-valued stochastic processes with time index ranging over the nonnegative real numbers. Then, semimartingales have historically been defined as follows.
Definition.
A semimartingale is a cadlag adapted process having the decomposition for a local martingale and a finite variation process .
More recently, the following alternative definition has also become common. For simple predictable integrands , the stochastic integral is easily defined for any process . The following definition characterizes semimartingales as processes for which this integral is well behaved.
Definition.
Writing for the supremum norm of a process , this definition characterizes semimartingales as processes for which
in probability as for each , where is any sequence of simple predictable processes satisfying . This property is necessary and, as it turns out, sufficient for the development of a theory of stochastic integration for which results such as bounded convergence holds.
The equivalence of these two definitions of semimartingales is stated by the Bichteler-Dellacherie theorem.
A stochastic process taking values in is said to be a semimartingale if is a semimartingale for each .
Title | semimartingale |
---|---|
Canonical name | Semimartingale |
Date of creation | 2013-03-22 18:36:38 |
Last modified on | 2013-03-22 18:36:38 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 8 |
Author | gel (22282) |
Entry type | Definition |
Classification | msc 60G07 |
Classification | msc 60G48 |
Classification | msc 60H05 |