Stochastic integration lecture 2
Stopping times in continuous time
Measurability with stopping times
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Stopping times in continuous time
Definition. A random variable T : Ω → R+ ∪ {∞} is said to be a
stopping time if (T ≤ t) ∈ Ft for all t ≥ 0.
Important difference between discrete and continuous time:
• In discrete time, a variable T : Ω → N ∪ {∞} is a stopping time if
and only if (T ≤ n) ∈ Fn for n ≥ 1, and if and only if (T = n) ∈ Fn
for n ≥ 1.
• In continuous time, T : Ω → R+ ∪ {∞} is a stopping time if and only
if (T ≤ t) ∈ Ft for t ≥ 0, but not if and only if (T = t) ∈ Ft for
t ≥ 0.
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Stopping times in continuous time
Let T be a stopping time.
• T is finite if T maps into R+ .
• T is bounded if T maps into a bounded subset of R+ .
Definition. For T a stopping time, we define
FT = {F ∈ F|F ∩ (T ≤ t) ∈ Ft for t ≥ 0}.
FT is a σ-algebra. We refer to FT as the σ-algebra of events determined
at T . If T is a constant, say s, the σ-algebra of events determined at T is
equal to Fs .
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Stopping times in continuous time
Lemma 2.1.7. The following holds.
1
Any constant in [0, ∞] is a stopping time.
2
T ≥ 0 is a stopping time if and only if (T < t) ∈ Ft , t ≥ 0.
3
If S and T are stopping times, so are S ∧ T , S ∨ T and S + T .
4
Let F ∈ FT with T a stopping time. T 1F + ∞1F c is a stopping time.
5
If S ≤ T with S and T stopping times, then FS ⊆ FT .
Lemma 2.1.8. Let (Tn ) be a sequence of stopping times. Then supn Tn
and inf n Tn are stopping times as well.
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Stopping times in continuous time
If (Xn )n≥1 is an adapted discrete-time process and B ∈ B, we can put
T = inf{n ≥ 1 | Xn ∈ B} and obtain (T ≤ n) = ∪nk=1 (Xk ∈ B) ∈ Fn ,
showing that T is a discrete-time stopping time.
In continuous time, the same result holds, given that X is continuous and
adapted. This is a special case of the début theorem. This illustrates
significant differences between discrete and continuous time:
• The proof of the stopping time property of “first hitting times” in
discrete time is not very hard.
• The proof of the début theorem is very hard.
• The proof makes use of the usual conditions in a nontrivial manner.
We will show the continuous-time result in the simpler case where B is
open or closed.
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Stopping times in continuous time
Example. Let X be a Brownian motion and T = inf{t ≥ 0 | Xt = 2}.
2
1
0
-1
Value of process
3
4
First hitting time of B, B = { 2 }
0
2
4T
6
8
10
t
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Stopping times in continuous time
Lemma 2.1.9. Let X be a continuous adapted process, and let U be an
open set in R. Let T = inf{t ≥ 0 | Xt ∈ U}. Then T is a stopping time.
Proof. By continuity, we obtain
(T < t) = ∪s∈Q+ ,s<t (Xs ∈ U),
Lemma 2.1.7(2) now yields the result.
Lemma 2.1.10. Let X be a continuous adapted process, and let F be a
closed set in R. Let T = inf{t ≥ 0 | Xt ∈ F }. Then T is a stopping time.
Proof. This follows as closed sets can be obtained as particular decreasing
countable intersections of open sets.
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Stopping times in continuous time
Definition. Let X be a process and T a stopping time. The process X
stopped at T , denoted X T , is defined by XtT = XT ∧t .
We say that X has initial value zero if X0 is zero. We say that X is
bounded by c if |Xt | ≤ c for all t ≥ 0.
Lemma 2.1.11. Let X be a continuous adapted process with initial value
zero. Define Tn = inf{t ≥ 0 | |Xt | > n}. Then (Tn ) is a sequence of
stopping times increasing to infinity, and the process X Tn is bounded by n.
Proof. Apply continuity of X .
Lemma 2.1.11 shows that when considering the process only on the
“stochastic interval” of time up to Tn , the process is bounded.
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Stopping times in continuous time
Example. Let X be a Brownian motion and T2 = inf{t ≥ 0 | Xt > 2}.
Process stopped at first hitting time of ( 2 ,\infty )
2
0
-2
Value of process
4
6
Process X
Stopped process X^T
0
2
4
T
6
8
10
t
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Measurability with stopping times
If X is adapted, we know by definition that Xt is Ft measurable for t ≥ 0.
Question: If T is a finite stopping time, is XT then FT measurable?
Answer: Not if we only assume that X is adapted. However, in the case
where X is progressive, the result holds.
This is an example of how progressive measurability, and not adaptedness,
is the “correct” criterion for ensuring that a stochastic process works
property together with the filtration.
Lemma 2.1.12. Let X be progressively measurable, and let T be a
stopping time. Then XT 1(T <∞) is FT measurable and X T is progressively
measurable.
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Measurability with stopping times
Lemma 2.1.13. Let T and S be stopping times. Assume that Z is FS
measurable. Then Z 1(S<T ) and Z 1(S≤T ) are FS∧T measurable.
Informally, the result states that at the time S ∧ T , it is possible to know
whether S is less or strictly less than T or not, and in the affirmative case,
variables depending only on information up to time S are known as well.
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