View source for Stopping time

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903802.png" />, be a non-decreasing family of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903803.png" />-algebras on a [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903805.png" /> is an interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903806.png" /> or a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903807.png" />. Then a stopping time (relative to this family of subalgebras) is a mapping ( a [[Random variable|random variable]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903808.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s0903809.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038010.png" />. Such a random variable is also called an optional random variable. This condition has the interpretation that the (time-valued) random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038011.png" /> has no knowledge of the future, since the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038012.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038013.png" /> embodies  "random events up to time t" . Many stopping times arise as  "the point of time at which a given random event is observed for the first time" ; for instance, the first time of entry of a stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038014.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090380/s09038015.png" /> (hitting time). In the (translated) Russian literature the phrase [[Markov moment|Markov moment]], or Markov time, is often used for stopping time. Occasionally one also finds the phrase non-anticipating time. Stopping times naturally arise, e.g., in optimal stopping problems, cf., e.g., [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,   "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. 332  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lamperti,   "Stochastic processes" , Springer  (1977)  pp. 210–213</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.L. Chung,   "Elementary probability theory with stochastic processes" , Springer  (1974)  pp. 269</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "Controlled stochastic processes" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.M. Rao,   "Stochastic processes and integration" , Sijthoff &amp; Noordhoff  (1979)</TD></TR></table>