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 ${\displaystyle ({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$

${\displaystyle X_{n}{\text{ ist }}{\mathcal {F}}_{n-1}{\text{-messbar }}}$

for everyone , the process is called predictable , previsible or predictable . ${\displaystyle n\in \mathbb {N} }$

Steady fall

In the continuous-time case, one defines the predictable σ-algebra on a ${\displaystyle \Omega \times [0,\infty )}$

${\displaystyle {\mathcal {P}}:=\sigma (X\,|\,X{\text{ ist adaptierter linksstetiger Prozess }})}$

(see adapted stochastic process , left-continuous process ). A process is said to be predictable if there is a measurable image. ${\displaystyle (\omega ,t)\mapsto X_{t}(\omega )}$ ${\displaystyle {\mathcal {P}}}$

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 ${\displaystyle {\mathcal {F}}_{n-1}}$ ${\displaystyle X_{n}}$ ${\displaystyle {\mathcal {F}}_{n-1}}$

${\displaystyle \operatorname {E} (X_{n}|{\mathcal {F}}_{n-1})=X_{n}}$.

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. ${\displaystyle X_{n}}$ ${\displaystyle {\mathcal {F}}_{n-1}}$ ${\displaystyle \sigma (X_{n})\subset {\mathcal {F}}_{n-1}}$

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 .