Completed martingale
A closed martingale is a special martingale in probability theory and thus a stochastic process . Those martingales that have a final element are clearly completed. Such martingales already converge on the basis of the properties assigned to them via the definition. Conversely, the question of whether a martingale is closed or can be closed by a random variable can be interpreted as a question of the convergence of the martingale .
definition
Consider a martingale with regard to filtration
It means a closed martingale if there is a and a are such that for all
- and
applies.
Is a submartingale so is analogous to complete when there is a and a are such that for all
- and
applies.
properties
Every completed martingale is always a doob martingale , so it can be in the form
represent for an integrable random variable . In this specific case , the random variable is the last element of the martingale.
Conversely, every Doob martingale can also be closed by putting as well as and , the σ-algebra of the underlying probability space.
In addition, closed martingales as well as closed, nonnegative submartingales can always be integrated to the same degree and thus converge almost certainly and in the first mean .
Conversely, according to the martingale convergence theorem, for every equally integrable martingale there is a random variable that can be measured with respect to
is so that and complete the martingale. It is the limit in the first agent and the almost sure convergence.
literature
- Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 272-275 , doi : 10.1007 / 978-3-642-45387-8 .