Predictable process
A predictable process , also called a predictable process , a previsible process or a predictable process , is a special stochastic process in which it is possible to look a short time step into the future. This does not mean that outputs are already known, but only that information can be obtained about the distribution. Predictable processes play a role, for example, in the Doob decomposition , which decomposes any integrable stochastic process into two sub-processes in discrete time: a martingale and a predictable process. They are also used in the definition of the discrete stochastic integral and the stochastic integral .
definition
Discreet case
Let there be a filtration and a stochastic process . Always applies
for everyone , the process is called predictable , previsible or predictable .
Steady fall
In the continuous-time case, one defines the predictable σ-algebra on a
(see adapted stochastic process , left-continuous process ). A process is said to be predictable if there is a measurable image.
Interpretation of the discrete case
The σ-algebra models the information that is available at time n-1 . If one now considers the conditional expectation of the random variable , taking into account the fact that the information from is already available, so is
- .
This follows from the fact that it is measurable and therefore . If the information from the ( n-1 ) th step is available, everything can already be said about the outputs in the n th step.
example
- Any process provided with the filtering of the complete information is predictable.
- Simple examples of time-continuous predictable processes are the elementary predictable stochastic processes .
Web links
- AN Shiryaev: Predictable random process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- AN Shiryaev: Predictable sigma-algebra . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
literature
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
- Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .