Weak Markov property

The weak Markov property is a property of a stochastic process in probability theory . It is used to define general Markov processes and is a tightening of the elementary Markov property , since, in contrast to this, it still requires that the transition probabilities between the states are independent of the point in time of the transition. Usually the weak Markov property is referred to as “the Markov property” and the addition “weak” is omitted.

definition

A stochastic process with values ​​in and amount of time is given , which is also closed with regard to addition and which contains 0. Let the produced filtration of the process. ${\displaystyle (E,{\mathcal {B}}(E))}$ ${\displaystyle T\subset [0,\infty )}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$

The Markov kernel of the transition probabilities to the time difference is defined as the kernel from to through ${\displaystyle t}$ ${\displaystyle (E,{\mathcal {B}}(E))}$ ${\displaystyle (E,{\mathcal {B}}(E))}$

${\displaystyle \kappa _{t}(x,A):=P_{x}(X_{t}\in A)}$

for . Here is the probability of being in when you started in. ${\displaystyle A\in {\mathcal {B}}(E)}$ ${\displaystyle P_{x}(X_{t}\in A)}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle X_{0}=x\in E}$

The process then has the weak Markov property if that holds for anything and all and all ${\displaystyle s,t\in T}$ ${\displaystyle A\in {\mathcal {B}}(E)}$ ${\displaystyle x\in E}$

${\displaystyle P_{x}(X_{s+t}\in A|{\mathcal {F}}_{s})=\kappa _{t}(X_{s},A)}$

is ( -almost sure). ${\displaystyle P_{x}}$

interpretation

The filtering contains the information about the course of the process from the start to the point in time ; accordingly, according to the conditional expected value, the conditional probability is the probability of being in at a later point in time if the prior knowledge of the process is known. ${\displaystyle ({\mathcal {F}}_{s})}$ ${\displaystyle s}$ ${\displaystyle P(X_{s+t}\in A|{\mathcal {F}}_{s})}$ ${\displaystyle s+t}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {F}}_{s}}$

According to the above description, the probability of being in after time units is then at the start in . This means the following: If you fix a state at any point in time and then go back in time steps from this state with the knowledge of the entire past of the process , the probability of the occurrence of the event is the same as if you were directly in the fixed state started and moved forward. The past of the process therefore has no influence on the transition probabilities. In this way the process has a “short memory”. In addition, the point in time has no influence on the transition probabilities, so the process is homogeneous. ${\displaystyle \kappa _{t}(X_{s},A)}$ ${\displaystyle X_{s}}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle s}$ ${\displaystyle X_{s}}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle X_{s}}$ ${\displaystyle t}$ ${\displaystyle s}$

Generalizations

A generalization of the weak Markov property is the strong Markov property . In a Markov process , it requires that the weak Markov property not only applies to deterministic points in time, but that it also applies to (random) stop times .

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 , doi : 10.1007 / 978-3-642-36018-3 .