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_{now}_{fut}_{1} and Filt(2)=Ω_{fut}_{2}. From a., b. and c. we know that:

d. {w, where w is Element of Ω: ST.w=0} in Ω_{now}

{w, where w is Element of Ω: ST.w=1} in Ω_{fut}_{1} and

{w, where w is Element of Ω: ST.w=2} in Ω_{fut}_{2}.

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 _{2}(1) which is equal to 80. The same argumentation can be applied 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 _{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 _{3}(4) which is equal to 120.

In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [

#### Keywords

- stopping time
- stochastic process

#### MSC 2010

- 60G40
- 03B35

Duality Notions in Real Projective Plane Finite Dimensional Real Normed Spaces are Proper Metric Spaces Relationship between the Riemann and Lebesgue Integrals Improper Integral. Part I About Graph Sums Improper Integral. Part II Automatization of Ternary Boolean Algebras Prime Representing Polynomial Quadratic Extensions The 3-Fold Product Space of Real Normed Spaces and its Properties