We start with the definition of stopping time according to [
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=tomorrow,2=the day after tomorrow.
We can prove:
c. {w, where w is Element of Ω: ST.w=0}=∅ & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} and
ST is a stopping time.
We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ω
d. {w, where w is Element of Ω: ST.w=0} in Ω
{w, where w is Element of Ω: ST.w=1} in Ω
{w, where w is Element of Ω: ST.w=2} in Ω
The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [
As an interpretation for our installed functions consider the given adapted stochastic process in the article [
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
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
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
In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [