# Markov moment

Markov time; stopping time

2010 Mathematics Subject Classification: Primary: 60G40 [MSN][ZBL]

A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let $( \Omega , {\mathcal F} )$ be a measurable space with a non-decreasing family $( {\mathcal F} _ {t} )$, $t \in T$, of $\sigma$- algebras of ${\mathcal F}$( $T = [ 0 , \infty ]$ in the case of continuous time and $T = \{ 0 , 1 ,\dots \}$ in the case of discrete time). A random variable $\tau = \tau ( \omega )$ with values in $T \cup \{ + \infty \}$ is called a Markov moment or Markov time (relative to the family $( {\mathcal F} _ {t} )$, $t \in T$) if for each $t \in T$ the event $\{ \tau ( \omega ) \leq t \}$ belongs to ${\mathcal F} _ {t}$. In the case of discrete time this is equivalent to saying that for any $n \in \{ 0 , 1 ,\dots \}$ the event $\{ \tau ( \omega ) = n \}$ belongs to ${\mathcal F} _ {n}$.

## Contents

### Examples.

1) Let $X ( t)$, $t \in T$, be a real-valued right-continuous random process given on $( \Omega , {\mathcal F} )$, and let ${\mathcal F} _ {t} = \sigma \{ \omega : {X ( s), s \leq t } \}$. Then the random variables

$$\tau ( \omega ) = \inf \{ {t \geq 0 } : {X ( t) \in B } \}$$

and

$$\sigma ( \omega ) = \inf \{ {t > 0 } : {X ( t) \in B } \} ,$$

that is, the (first and first after $+ 0$) times of hitting the (Borel) set $B$, form Markov moments (in the case $\{ \cdot \} = \emptyset$ it is assumed that $\inf \emptyset = \infty$).

2) If $w ( t)$, $t \geq 0$, is a standard Wiener process, then the Markov moment

$$\tau = \inf \{ {t \geq 0 } : {w ( t) \geq a } \} ,\ a > 0 ,$$

has probability density

$$P ( t) = \ \frac{a}{t ^ {3/2} \sqrt {2 \pi } } e ^ {- a ^ {2} / 2 t } .$$

Here ${\mathsf P} \{ \tau < \infty \} = 1$, but ${\mathsf E} \tau = \infty$.

3) The random variable

$$\gamma = \inf \{ {t > 0 } : {X ( s) \in B , s \geq t } \} ,$$

being the first time after which $X _ {t}$ remains in $B$, is an example of a non-Markov moment (a random variable depending on the "future" ).

Using the idea of a Markov moment one can formulate the strong Markov property of Markov processes (cf. Markov process). Markov moments and stopping times (that is, finite Markov moments) play a major role in the general theory of random processes and statistical sequential analysis.

#### References

 [GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027