Elementary Markov property

The elementary Markov property is a property of stochastic processes in probability theory . It is a generally formulated condition for the extent to which the process is influenced by its past and enables the definition of Markov processes under diverse framework conditions.

definition

Given is an index set and a stochastic process with values ​​in and with the generated filtration . ${\displaystyle T\subset \mathbb {R} }$ ${\displaystyle X=(X_{t})_{t\in T}}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$

The process has the elementary Markov property if that applies to everyone and everyone with ${\displaystyle X}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle s,t\in T}$ ${\displaystyle s\leq t}$

${\displaystyle P(X_{t}\in A|{\mathcal {F}}_{s})=P(X_{t}\in A|\sigma (X_{s}))}$.

interpretation

Building on the conditional expected value , the term can be interpreted as the best prediction that can be given for the event if one has the information from . ${\displaystyle P(X\in A|{\mathcal {B}})}$ ${\displaystyle X\in A}$ ${\displaystyle {\mathcal {B}}}$

The filtration now contains all information about the course of the process from the beginning to the point in time , the σ-algebra only the information about the point in time . ${\displaystyle {\mathcal {F}}_{s}}$ ${\displaystyle s}$ ${\displaystyle \sigma (X_{s})}$ ${\displaystyle s}$

The elementary Markov property now states that the best prediction for an event does not change with the information situation. Regardless of whether you know the entire process up to or only the current state in , the forecast for the further course of the process is not changed. This is the “memorylessness” or “short memory” that characterizes all Markov processes. ${\displaystyle s}$ ${\displaystyle s}$

Relation to the weak Markov property

The elementary Markov property is more general than the weak Markov property . This requires the existence of a Markov nucleus that describes the transition probabilities. In addition, in contrast to the elementary Markov property, it requires that the transition probabilities are independent of time, so it is only fulfilled by homogeneous Markov processes.

Web links

• Markov Process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• TO Shiryaev: Markov Property . 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 .