Strong Markov quality

The strong Markov property is a property that a class of stochastic processes , more precisely Markov processes , can, but does not have to, have. It can therefore be assigned to probability theory . The strong Markov property is a tightening of the weak Markov property , in which a deterministic point in time is replaced by a (random) stop time .

definition

A Markov process with distributions and index set is given . ${\displaystyle (P_{x})_{x\in E}}$ ${\displaystyle T}$

The process now has the strong Markov property , if for every bounded, - -measurable function and for every finite stop time and all the equation ${\displaystyle {\mathcal {B}}(E)^{\otimes T}}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle f\colon E^{\times T}\to \mathbb {R} }$ ${\displaystyle \tau }$ ${\displaystyle x\in E}$

${\displaystyle \operatorname {E} _{x}(f((X_{\tau +t})_{t\in T})|{\mathcal {F}}_{\tau })=\operatorname {E} _{X_{\tau }}(f(X))}$

applies.

It is the σ-algebra of τ-past and defining ${\displaystyle {\mathcal {F}}_{\tau }}$

${\displaystyle \operatorname {E} _{X_{\tau }}(f(X)):=\int _{E^{\times T}}f(y)\kappa (X_{\tau },\mathrm {d} y)}$.

For countable index sets

If one denotes the distribution of the process at the start in , then the strong Markov property for countable index sets is equivalent to ${\displaystyle {\mathcal {L}}_{x}}$ ${\displaystyle x}$ ${\displaystyle T=\mathbb {N} }$

${\displaystyle {\mathcal {L}}_{x}((X_{\tau +t})_{t\in \mathbb {N} }|{\mathcal {F}}_{\tau })={\mathcal {L}}_{X_{\tau }}((X_{t})_{t\in \mathbb {N} })}$

for all finite stop times . In this case it can be proven that the strong Markov property already follows from the weak Markov property. ${\displaystyle \tau }$

Web links

• 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 , p. 362 , doi : 10.1007 / 978-3-642-36018-3 .