@MISC{FINANCE4.MIZ,
  AUTHOR = {Jaeger, Peter},
  TITLE = {Introduction to Stopping Time in Stochastic Finance Theory},
  SECTION1 = {Definition of Stopping Time},
  SECTION2 = {Examples of Stopping Times},
  DAY = {27},
  MONTH = {June},
  YEAR = {2017},
  ADDRESS1 = {},
  SUMMARY = {We start with the definition of stopping time according to
             (\cite{follmerschied:2004},p.283). We prove, that different
             defintions for stopping time can coincide. We give
             examples of stopping times using constant-functions or
             functions defined with the operator max or min (defined in
             \cite{klenke:2006},p.37-38). Last we give an example with
             some given filtration. Stopping times are very important for
             stochastic finance. A stopping time is the moment, where a
             certain event occurs \cite{kremer:2006},p.372), and can be
             used together with stochastic processes
             (\cite{follmerschied:2004},p.283).
             Look at the following example:
             We install a function ST:{1,2,3,4}->{0,1,2} \/ {+infty}
             We define:
              a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2
              b. The set {0,1,2} consists of time points:
                 0=now,1=tomorow,2=the day after tomorrow
             We can prove:
              b. {w where w is Element of Omega: ST.w=0}={} &
                 {w where w is Element of Omega: ST.w=1}={1,2} &
                 {w where w is Element of Omega: ST.w=2}={3,4}
              and
              c. ST is a stopping time.
             We use a function Filt as Filtration of {01,2},Sigma where
              Filt(0)=Omega_now, Filt(1)=Omega_fut1 and Filt(2)=Omega_fut2.
             From a.,b. and c. we know that:
              d. {w where w is Element of Omega: ST.w=0} in Omega_now and
                 {w where w is Element of Omega: ST.w=1} in Omega_fut1 and
                 {w where w is Element of Omega: ST.w=2} in Omega_fut2
             The sets in d. are events, which occur at the time points
             0(=now),1(=tomorrow) or 2(=the day after tomorrow), see also
              \cite{kremer:2006},p.371. Suppose ST(1)=+infty,
             then this means, that for ST(1) never stops.
             As an interpretation for our installed functions consider
             the given adapted stochastic process in the article FINANCE3.
             ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped
             in 1 (=tomorrow). That tells us, that we have to look at the
             value f_2(1) which is equal to 80. Same argumentation for the
             element 2 in {1,2,3,4}.
             ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped
             in 2 (=the day after tomorrow). That tells us, that we have to
             look at the value f_3(3) which is equal to 100.
             ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped
             in 2 (=the day after tomorrow). That tells us, that we have to
             look at the value f_3(4) which is equal to 120.
             In real world, these functions can be used for questions like:
             When does the share price exceed a certain limit, see
             \cite{kremer:2006},p.372).
}}

@book{follmerschied:2004,
  author={F\"ollmer, Hans and Schied, Alexander},
  title={Stochastic Finance: An Introduction in Discrete Time},
  edition={2nd},
  volume={27},
  series={Studies in Mathematics},
  publisher={de Gruyter},
  year={2004},
  ADDRESS={Berlin}}

@book{klenke:2006,
author={Klenke, Achim},
title={Wahrscheinlichkeitstheorie},
publisher={Springer-Verlag},
year={2006},
ADDRESS={Berlin, Heidelberg}}

@book{kremer:2006,
  author={Kremer, Jürgen},
  title={Einführung in die diskrete Finanzmathematik},
  edition={},
  publisher={Springer-Verlag},
  year={2006,
  ADDRESS={Berlin,Heidelberg,New York}}
