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.
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.
A semimartingale is a cadlag adapted process such that
is bounded in probability for each .
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 .
|Date of creation||2013-03-22 18:36:38|
|Last modified on||2013-03-22 18:36:38|
|Last modified by||gel (22282)|