Statistics and Probability Letters 83 (2013) 345–349

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

On simple representations of stopping times and stopping timesigma-algebrasTom Fischer ∗

Institute of Mathematics, University of Wuerzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074 Wuerzburg, Germany

a r t i c l e i n f o

Article history:Received 20 June 2012Received in revised form 25 September2012Accepted 27 September 2012Available online 3 October 2012

MSC:60G4097K5097K60

Keywords:Début TheoremHitting timeStopped sigma-algebraStopping timeStopping time sigma-algebra

a b s t r a c t

There exists a simple, didactically useful one-to-one relationship between stopping timesand adapted càdlàg (RCLL) processes that are non-increasing and take the values 0 and 1only. As a consequence, stopping times are always hitting times. Furthermore, we showhow minimal elements of a stopping time sigma-algebra can be expressed in terms ofthe minimal elements of the sigma-algebras of the underlying filtration. This facilitates anintuitive interpretation of stopping time sigma-algebras. A tree example finally illustrateshow these for students notoriously difficult concepts, stopping times and stopping timesigma-algebras, may be easier to grasp by means of our results.

© 2012 Elsevier B.V. All rights reserved.

1. Stopping times are hitting times: a natural representation

The possibly most popular example of a stopping time is the ‘hitting time’ of a set by a stochastic process, defined asthe first time at which a certain pre-specified set is hit by the process considered. Often this example is considered for aBorel-measurable set and a one-dimensional real-valued adapted process on a discrete time axis. Similar results in moregeneral setups are usually collectively called the Début Theorem; see Bass (2010, 2011) for a proof.

Astonishingly, it seems to be less widely taught (and maybe known) that the converse is true as well: for any stoppingtime there exists an adapted stochastic process and a Borel-measurable set such that the corresponding hitting time will beexactly this stopping time. Furthermore, the stochastic process can be chosen very intuitively: it will be 1 until just beforethe stopping time is reached, from which time on it will be 0. The 1–0-process therefore first hits the Borel set 0 at thestopping time. Hence, a one-to-one relationship is established between stopping times and adapted càdlàg processes thatare non-increasing and take the values 0 and 1 only.

A 1–0-process as described above is obviously akin to a random traffic lightwhich can go through three possible scenariosover time (here, ‘green’ stands for ‘1’ and ‘red’ stands for ‘0’): (i) it stays red forever (‘stopped immediately’); (ii) it is green atthe beginning, then turns red, and stays red for the rest of the time (‘stopped at some stage’); (iii) it stays green forever (‘neverstopped’). So, the traffic light can never change back to green once it has turned red (stopped once means stopped forever),and, for the adaptedness of the 1–0-process, it can only change on the basis of information up to the corresponding point in

∗ Tel.: +49 931 3188911.E-mail address: tom.fischer@uni-wuerzburg.de.

0167-7152/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2012.09.024

346 T. Fischer / Statistics and Probability Letters 83 (2013) 345–349

time. This very intuitive interpretation of a stopping time as the time when such a ‘traffic light’ changes is considerablyeasier to understand than the concept of a random time which is ‘known once it has been reached’—one of the verbalinterpretations of the usual standard definition of a stopping time (see Definition 2).

While this representation and alternative definition of stopping times seems natural and didactically useful, it does notseem to be widely taught, or otherwise one would expect to find it in textbooks. However, there is no mention of it instandard textbooks on probability such as Billingsley (1995) and Bauer (2002) (Bauer as an example of a popular Germanstochastics textbook), or in standard textbooks on stochastic processes, stochastic calculus and stochastic mathematicalfinance such as Karatzas and Shreve (1991) and Bingham and Kiesel (2004).

Definition 1. We define the time axis as a set T ⊂ R such that if h ∈ R is an accumulation point of T and h < supT, thenh ∈ T.

The above requirement for the time axis (where we allow supT = +∞) is very mild, as in most applications evenlyspaced discrete points in time or continuous time intervals would be considered, and the required condition would befulfilled.

Definition 2. A stopping timewith respect to a filtrationF = (Ft)t∈T on a probability space (Ω, F∞, P) is a randomvariablewith values in T ∪ +∞ such that

τ ≤ t ∈ Ft (t ∈ T). (1)

Interpreted verbally, (1) means that at any time t ∈ T one knows – on the basis of information up to time t – whetherone has been stopped already, or not. Note that Ft ⊂ F∞ is assumed for t ∈ T.

In the following, we call a real-valued stochastic process càdlàg if for all ω ∈ Ω the paths X·(ω) : T → R have theproperty of being right-continuous with left-handed limits (RCLL). Note that for right-continuity to be well-defined, thecondition of Definition 1 must hold on T in order to ensure that any point h ∈ R to which a sequence in T might convergefrom the right is indeed contained in T and, hence, to ensure that Xh(ω) exists.

Definition 3. We call an adapted càdlàg process X = (Xt)t∈T on (Ω, F∞, F, P) a ‘stopping process’ if

Xt(ω) ∈ 0, 1 (ω ∈ Ω, t ∈ T) (2)

and

Xs ≥ Xt (s ≤ t; s, t ∈ T). (3)

Obviously, Xt = 1 ∈ Ft for t ∈ T. For a finite or infinite discrete time axis given by T = tk : k ∈ N, tk ≥ tj if k ≥ j,an adapted process fulfilling (2) and (3) is automatically càdlàg. This follows from the observation that in this case T isbounded from below and either (a) is finite as a set, or (b) is infinite and has one accumulation point, supT, which liesoutside T, and therefore is bounded, or (c) has no accumulation point and is unbounded from above. Evidently, such T alsofulfills Definition 1.

Definition 4. For a stopping process X on (Ω, F∞, F, P), define

τ X (ω) =

+∞ if Xt(ω) = 1 for all t ∈ T,mint ∈ T : Xt(ω) = 0 otherwise. (4)

The minimum in the lower case exists because of the càdlàg property for each path. By definition of τ X , it is clear that

Xt = 1τX>t (t ∈ T), (5)

which by adaptedness of X and τ X > t = τ X≤ tC implies

τ X≤ t ∈ Ft (t ∈ T). (6)

Therefore, τ X is a stopping time for any stopping process X . Clearly, τ X is the first time of X hitting the Lebesgue-measurableset 0.

Definition 5. For a stopping time τ , define a stochastic process X τ= (X τ

t )t∈T by

X τt = 1τ>t (t ∈ T). (7)

By τ > t = τ ≤ tC and (1), X τ is an adapted process. One has X τs = 1τ>s ≥ 1τ>t = X τ

t for s ≤ t . Becauselimt↓τ(ω) X τ

t (ω) = 0 = X ττ(ω), X τ is càdlàg and hence a stopping process.

T. Fischer / Statistics and Probability Letters 83 (2013) 345–349 347

Theorem 1. The mapping

f : X −→ τ X (8)

is a bijection between the stopping processes and the stopping times on (Ω, F∞, F, P) such that

f −1: τ −→ X τ , (9)

and hence

τ Xτ= τ and X τX

= X . (10)

Proof. That f maps stopping processes X to stopping times τ X was seen in (6). That different stopping processes lead todifferent stopping times under f is obvious from (4); f is therefore an injection. For any stopping time τ , one has by (4) and(7) that

τ Xτ(ω) =

+∞ if 1τ>t(ω) = 1 for all t ∈ T,mint ∈ T : 1τ>t(ω) = 0 otherwise. (11)

Since 1τ>t(ω) = 0 if and only if τ(ω) ≤ t , the right-hand side of (11) is τ(ω). Therefore, τ(ω) = τ Xτ(ω), and f is a

surjection and therefore is a bijection. From (5) and (7), we obtain X τXt = 1τX>t = Xt for t ∈ T, which proves (9).

Note that there was no identification of almost surely identical stopping times or stopping processes in Theorem 1 or anyof the definitions; however, one can obviously transfer all results to equivalence classes of almost surely identical objects.

2. Minimal elements of stopping time sigma-algebras

Another concept that students of stochastic processes and probability usually struggle with is that of the so-calledstopping time σ -algebras.

Definition 6. Let τ be a stopping time on a filtered probability space (Ω, F∞, F, P). We define the stopping time σ -algebrawith respect to τ as

Fτ = F ∈ F∞ : F ∩ τ ≤ t ∈ Ft for all t ∈ T. (12)

It is well-known and straightforward to show that Fτ is indeed a σ -algebra.

Definition 7. For a measurable space (Ω, F ), we define the set of the minimal elements in the σ -algebra F by

A(F ) = A ∈ F : A = ∅, and if F ∈ F and F ⊂ A, then F = A. (13)

Eq. (13) means that A ∈ A(F ) cannot be ‘split’ in F , which is why elements of A(F ) are also referred to as ‘atoms’ ofF . Obviously, A(F ) ⊂ F . For |F | < +∞, it is therefore easy to see that A(F ) is a partition of Ω and that

F = σ(A(F )) (14)

since any non-empty F ∈ F can be written as a finite union of elements in A(F ).

Definition 8. For t ∈ T ∪ +∞, we denote the set of minimal elements in Ft by At = A(Ft). Further, we define

Atτ = A ∈ At : A ⊂ τ = t (t ∈ T ∪ +∞), (15)

Aτ =

t∈T∪+∞

Atτ . (16)

Note that the Atτ are disjoint for t ∈ T ∪ +∞.

Theorem 2. The elements of Aτ are minimal elements of Fτ , i.e. Aτ ⊂ A(Fτ ). If |F∞| < +∞, then Aτ = A(Fτ ) andFτ = σ(Aτ ).

Proof. (i) Aτ ⊂ Fτ : Let A ∈ Aτ , so, for some s ∈ T ∪ +∞, A ∈ As and A ⊂ τ = s and therefore A ∩ τ = s = A.Assume now that t < s for some t ∈ T. Then A ∩ τ ≤ t = A ∩ τ = s ∩ τ ≤ t = ∅ ∈ Ft . For t ∈ T, assume nowthat t ≥ s. Then A ∩ τ ≤ t = A ∩ τ = s ∩ τ ≤ t = A ∩ τ = s = A ∈ Fs ⊂ Ft . Hence, A ∩ τ ≤ t ∈ Ft for

348 T. Fischer / Statistics and Probability Letters 83 (2013) 345–349

Fig. 1. Example.

all t ∈ T, and therefore Aτ ⊂ Fτ . (ii) A ∈ Aτ , F ∈ Fτ and F ⊂ A implies F = A: (a) Assume that A ∈ A∞τ and F ∈ Fτ

with F ⊂ A. Clearly, A ∈ A∞, implying F = A since F ∈ F∞. (b) Assume that A ∈ Atτ for some t ∈ T and F ∈ Fτ with

F ⊂ A. Therefore, A ∈ At and F ⊂ A ⊂ τ = t, and hence F ∩ τ = t = F . As F ∈ Fτ , one has F ∩ τ ≤ t ∈ Ft . Sinceτ = t ∈ Ft , F ∩ τ ≤ t ∩ τ = t = F ∩ τ = t = F ∈ Ft , but A ∈ At , and therefore F = A. (i) and (ii) prove the firststatement of the theorem. Assume now that |F∞| < +∞. Aτ is then a partition of Ω , because any two distinct sets in Aτ

are disjoint, and, since τ = t ∈ Ft for t ∈ T ∪ +∞, one has

Atτ = τ = t and, hence,

Aτ = Ω . This proves the

second statement. The last statement follows from (14).

The following result is well-known.

Proposition 1. If |F∞| < +∞, then σ(τ) ⊂ Fτ and, in general, σ(τ) = Fτ .

Proof. τ = t (t ∈ T) and τ = +∞ are the minimal elements of σ(τ) if they are non-empty, but it is well-known andstraightforward to see that these sets are elements of Fτ , too. Therefore, σ(τ) ⊂ Fτ . It is an easy exercise to find exampleswhere σ(τ) = Fτ (see the example below).

We can interpret the filtered probability space (Ω, F∞, F, P) as a random ‘experiment’ or ‘experience’ that develops overtime. The common interpretation of the filtration F = (Ft)t∈T is that, if it is possible to repeat the experiment arbitrarilyoften until time t ∈ T, the maximum information that can be obtained about the experiment is Ft . Hence, Ft represents(potentially) available information up to time t .

Using Theorem 2 for |F∞| < +∞, we can interpret the stopping time σ -algebra Fτ as the maximum information thatcan be obtained from repeatedly carrying out the experiment up to the random time τ . This is straightforward from thedefinition of the At

τ in (15). While this interpretation is, of course, generally known, minimal sets are usually not used toderive it. However, the example belowwill illustrate that this is a very natural way of interpreting stopping time σ -algebras.

3. An example

In Fig. 1, we see the usual interpretation of a discrete time finite space filtration as a stochastic tree. In such a setting,the set of paths of maximal length represents Ω . In this case, Ω = ω1, . . . , ω8. A path up to some node at time t (here

T. Fischer / Statistics and Probability Letters 83 (2013) 345–349 349

T = 0, 1, 2, 3 and F3 = F∞ = P (Ω)) represents the set of those paths of full length that have this path up to time t incommon. The paths up to time t represent the minimal elements (atoms) At of Ft . For instance, in the example of Fig. 1,

A1 = ω1, ω2, ω3, ω4, ω5, . . . , ω8. (17)

A stopping time τ (see Fig. 1) and the corresponding stopping process X τ are given (X τ is given by the values of the nodes ofthe tree). The intuitive interpretation of the stopping time as the random time at which X τ jumps to (‘hits’) 0 becomes clear(see the boxed zeros). Furthermore, the paths up to those boxed zeros represent the minimal elements Aτ of the stoppingtime σ -algebra Fτ . This follows of course from the fact that for any A ∈ At

τ one has A ⊂ τ = t by (15) and thereforeX τt (A) = 0, but X τ

s (A) = 1 for s < t . So, in a tree example such as the given one, the ‘frontier of first zeros’ (or, moreprecisely, the paths leading up to it) describes the stopping time σ -algebra (as well as the stopping time itself). Therefore,it becomes very obvious in what sense Fτ contains the information in the system that can be obtained up to time τ . In thecase of the example,

Aτ = ω1, ω2, ω3, ω4, ω5, ω6, ω7, ω8. (18)

It is also easy to see that in this case the minimal elements of σ(τ), denoted by A(σ (τ )), are given by

A(σ (τ )) = ω1, ω2, ω5, ω6, ω3, ω4, ω7, ω8. (19)

Therefore, Aτ = A(σ (τ )) and, hence, σ(τ) = Fτ , as stated in Proposition 1.

References

Bass, R.F., 2010. The measurability of hitting times. Electronic Communications in Probability 15, 99–105.Bass, R.F., 2011. Correction to ‘‘the measurability of hitting times’’. Electronic Communications in Probability 16, 189–191.Bauer, H., 2002. Wahrscheinlichkeitstheorie. 5. Auflage. de Gruyter.Billingsley, P., 1995. Probability and Measure, third ed. Wiley-Interscience.Bingham, N.H., Kiesel, R., 2004. Risk-Neutral Valuation. Pricing and Hedging of Financial Derivatives, second ed. Springer.Karatzas, I., Shreve, S.E., 1991. Brownian Motion and Stochastic Calculus, second ed. Springer.