Section 6.5. Stopping times.
Let 𝑋𝑡 , 𝑡 ∈ 𝑇 ⊆ ℝ, be a stochastic process (it will be our process of reference). A
random variable 𝜏 = 𝜏 (𝜔) taking values in 𝑇 ∪ {∞} (that is, 𝜏 ∈ 𝑇 or is possibly infinite)
is called a stopping time if for every 𝑡 ∈ 𝑇 the event {𝜏 ≤ 𝑡} is determined by the values
of our stochastic process 𝑋𝑢 for 𝑢 ≤ 𝑡: there exists a subset 𝐶 in the space of functions
𝑥𝑢 , 𝑢 ≤ 𝑡, such that
{ (
)
}
(6.5.1)
{𝜔 : 𝜏 (𝜔) ≤ 𝑡} = 𝜔 : 𝑋𝑢 (𝜔), 𝑢 ≤ 𝑡 ∈ 𝐶 .
That is, for every 𝑡 we should be able, observing the reference process 𝑋𝑢 up to this time,
to determine whether the moment 𝜏 has already come (𝜏 ≤ 𝑡) or not.
The name “stopping time” is used because of the use of this concept in the theory of
martingales: we’ll see that if we stop a martingale at a stopping random time 𝜏 , it still
remains a martingale. There is another name for the same class of mathematical objects:
Markov times: from their use in the theory of Markov processes.
The concept of stopping time has nothing to do with any probabilities or expectations;
so it does not belong to the theory of stochastic processes properly speaking but rather to
the set-theoretic introduction to it.
Example 6.5.1. Every constant 𝑡∗ ∈ 𝑇 , and also 𝜏 ≡ ∞ is a stopping time.
Indeed,
{
Ω
if 𝑡∗ ≤ 𝑡,
{𝜔 : 𝑡∗ ≤ 𝑡} =
(6.5.2)
∅
if 𝑡∗ > 𝑡.
{ (
)
}
Each of these events is represented as 𝜔 : 𝑋𝑢 (𝜔), 𝑢 ≤ 𝑡 ∈ 𝐶 with 𝐶 being some subset
in the space of functions 𝑥𝑢 , 𝑢 ≤ 𝑡: the event Ω, with 𝐶 being the whole space of these
functions, with the impossible event ∅, with 𝐶 = ∅.
As for 𝜏 = ∞, clearly for every real 𝑡
{𝜔 : ∞ ≤ 𝑡} = ∅.
(6.5.3)
Indeed we know by time 𝑡 whether 𝑡∗ has already come (and the time +∞ definitely
has not come by time 𝑡).
Example 6.5.2. Let 𝑡1 < 𝑡2 be two time points belonging to 𝑇 ; and let 𝐶 be a
subset of the space of functions 𝑥𝑢 , 𝑢 ≤ 𝑡1 . Take
{
𝑡1
if (𝑋𝑢 , 𝑢 ≤ 𝑡1 ) ∈ 𝐶,
(6.5.4)
𝜏=
𝑡2
otherwise.
Then 𝜏 is a stopping time.
Indeed,
⎧
∅



⎨ {(𝑋 , 𝑢 ≤ 𝑡 ) ∈ 𝐶}
𝑢
1
{𝜏 ≤ 𝑡} =

{(𝑋𝑢 , 𝑢 ≤ 𝑡) ∈ 𝐶𝑡 }


⎩
Ω
1
for 𝑡 < 𝑡1 ,
for 𝑡 = 𝑡1 ,
for 𝑡1 < 𝑡 < 𝑡2 ,
for 𝑡 ≥ 𝑡2 ,
(6.5.5)
where the set 𝐶𝑡 in the space of all functions 𝑥𝑢 , 𝑢 ≤ 𝑡, consists of all functions whose
part up to time 𝑡1 belongs to the set 𝐶.
Example 6.5.3. Let 𝑇 = {0, 1, 2, ..., 𝑛, ...}, and let 𝑋𝑡 , 𝑡 ∈ 𝑇 , be a process taking
values in a space SP. Let 𝐴 be a subset of SP. Then 𝜏 defined as the first time for which
𝑋𝑡 ∈ 𝐴 (the first reaching time) is a stopping time.
Only what are we to do if 𝑋𝑡 never reaches 𝐴? and there is no first moment of
reaching it? Let us take, in this case, 𝜏 = ∞. So the precise definition:
{
min{𝑡 : 𝑋𝑡 ∈ 𝐴}
if there are such 𝑡,
𝜏=
(6.5.6)
+∞
if there is no such 𝑡.
This seems to coincide with the way we would use the expression “plus infinity” in our everyday life:
if we tell somebody that something will happen at time +∞, in all probability we would mean that it will
never occur.
Let us check that this 𝜏 is a stopping time. For 𝑡 ∈ 𝑇 we have:
{𝜏 ≤ 𝑡} = {𝑋0 ∈ 𝐴} ∪ {𝑋1 ∈ 𝐴} ∪ ... ∪ {𝑋𝑡 ∈ 𝐴} = {(𝑋𝑢 , 0 ≤ 𝑢 ≤ 𝑡) ∈ 𝐶},
(6.5.7)
where the set 𝐶 ⊆ SP𝑡+1 is defined as the set of all sequences (𝑥0 , 𝑥1 , ..., 𝑥𝑡 ) for which at
least one element 𝑥𝑖 ∈ 𝐴.
Example 6.5.4. It the same situation let us consider the last time that 𝑋𝑡 ∈ 𝐴:
{
sup{𝑡 : 𝑋𝑡 ∈ 𝐴}
if there are such 𝑡,
(6.5.8)
𝜎=
and we don’t know what
if there is no such 𝑡.
The supremum is used here rather than maximum because the set of all 𝑡’s for which
𝑋𝑡 ∈ 𝐴 may be infinite (and then the supremum is equal to +∞). As for the case of
no such 𝑡’s, in order not to have to think of it, let us restrict ourselves to the case when
𝑋0 (𝜔) ∈ 𝐴 for all 𝜔 ∈ Ω; also let us suppose for simplicity that the number of visits of
𝑋𝑡 (𝜔) to the set 𝐴 is finite for every 𝜔.
It turns out that, in general, 𝜎 is not a stopping time.
Indeed, say, for 𝑡 = 3, suppose 𝑋0 ∈ 𝐴, 𝑋1 ∈
/ 𝐴, and 𝑋2 , 𝑋3 ∈ 𝐴. If the process 𝑋𝑡
never visits the set 𝐴 after time 𝑡 = 3, we have 𝜎 = 3; but if it does, 𝜎 > 3. So, observing
the process 𝑋𝑡 up to time 3 we cannot be sure whether 𝜎 ≤ 3 or > 3.
It may, of course, happen that because of something special about the process 𝑋𝑡 , the
random variable 𝜎 happens to be a stopping time; but generally speaking, no.
Example 6.5.3′ . 𝑇 = [0, ∞), 𝑋𝑡 , 𝑡 ≥ 0, is a process with continuous trajectories,
and 𝐴 is a closed subset of the space SP. Then the random variable 𝜏 defined by (6.5.6)
is a stopping time.
First of all, the minimum does exist if there are 𝑡’s with 𝑋𝑡 ∈ 𝐴: because of the
continuity of 𝑋𝑡 (𝜔) and closedness of the set 𝐴. If the set 𝐴 were not closed, say, if it were
an open set, we could speak only of the infimum inf{𝑡 : 𝑋𝑡 ∈ 𝐴}.
For 𝑡 ∈ [0, ∞) we have:
{𝜏 ≤ 𝑡} = {(𝑋𝑢 , 0 ≤ 𝑢 ≤ 𝑡) ∈ 𝐶},
2
(6.5.9)
where the set 𝐶 consists of all functions 𝑥𝑢 , 0 ≤ 𝑢 ≤ 𝑡, taking a value in the set 𝐴 for at
least one 𝑢 ∈ [0, 𝑡] (if we have observed 𝑋𝑢 for 𝑢 ∈ [0, 𝑡], we just look at this function:
if it reaches the set 𝐴 for one of these 𝑢’s, the time 𝜏 has come by the time 𝑡; if not, it
hasn’t).
Example 6.5.3′′ . 𝑇 = [0, ∞), 𝑋𝑡 , 𝑡 ∈ [0, ∞), is a real-valued process with continuous
trajectories, 𝐴 = (0, ∞) (an open, not closed set). Take
{
𝜏=
inf{𝑡 : 𝑋𝑡 ∈ 𝐴}
+∞
if there are such 𝑡,
if there is no such 𝑡.
(6.5.10)
It turns out that, generally, this is not a stopping time.
Indeed, suppose we observed the process 𝑋𝑡 for 0 ≤ 𝑡 ≤ 2, and got the following
realization:
{
− 2 + 𝑡2 ,
0 ≤ 𝑡 ≤ 1,
𝑋𝑡 (𝜔) =
(6.5.11)
2
− (2 − 𝑡) ,
1≤𝑡≤2
(make a picture of the graph, which is continuous; at 𝑡 = 1 both formulas yield the same
result). It may be that for this 𝜔 the trajectory will go up after 𝑡 = 2, e. g.:
𝑋𝑡 (𝜔) = (𝑡 − 2)2 ,
𝑡 ≥ 2;
(6.5.12)
for such an 𝜔 we have 𝜏 (𝜔) = inf(2, ∞) = 2. But it may be that the trajectory will go
down after having touched the level 0, e. g.:
𝑋𝑡 (𝜔) = −(𝑡 − 2)2 + (𝑡 − 2)3 ,
𝑡≥2
(6.5.13)
(make a picture of the graph). For 𝜔 for which (6.5.11), (6.5.13) hold, 𝜏 = 3.
So observing 𝑋𝑢 , 0 ≤ 𝑢 ≤ 2, we cannot decide whether the event {𝜏 ≤ 𝑡} has occurred
or not.
Not a stopping time.
Example 6.5.5. Let 𝜏 be an arbitrary stopping time; let us prove that 𝜎 = 𝜏 + 1
is also one (we suppose that the time parameter set 𝑇 is such that 𝑡 + 1 ∈ 𝑇 for 𝑡 ∈ 𝑇 ).
Let, for definiteness, 𝑇 = [0, ∞) or 𝑇 = {0, 1, 2, ..., 𝑛, ...}.
We have:
{
∅
if 𝑡 < 1,
(6.5.14)
{𝜎 ≤ 𝑡} = {𝜏 ≤ 𝑡 − 1} =
{(𝑋𝑢 , 0 ≤ 𝑢 ≤ 𝑡) ∈ 𝐷}
if 𝑡 ≥ 1,
where the set 𝐷 in the space of functions 𝑥𝑢 , 0 ≤ 𝑢 ≤ 𝑡, consists of all functions whose
restriction to the interval [0, 𝑡 − 1] belongs to the set 𝐶 used in the representation
{𝜏 ≤ 𝑡 − 1} = {(𝑋𝑢 , 0 ≤ 𝑢 ≤ 𝑡 − 1) ∈ 𝐶}.
(6.5.15)
Example 6.5.5′ . Let 𝑇 = [0, ∞), and let 𝜏 be a stopping time. The random
variable 𝜎 = 𝜏 /2 may not be a stopping time.
3
This is because the event
{𝜎 ≤ 𝑡} = {𝜏 ≤ 2𝑡} = {(𝑋𝑢 , 0 ≤ 𝑢 ≤ 2𝑡) ∈ 𝐶2𝑡 }
(6.5.16)
may be representable through the values of 𝑋𝑢 with 0 ≤ 𝑢 ≤ 𝒕, and may be not: one
cannot require you to stop the process at the half-time before it, say, reaches a set 𝐴 for
the first time.
Example 6.5.5′′ . Let 𝜏 be an arbitrary stopping time; and let ℎ(𝑡) be a nondecreasing left-continuous function on 𝑇 such that ℎ(𝑡) ≥ 𝑡 for every 𝑡 (make a picture of
the graph of such a function). Then the random variable
𝜎 = ℎ(𝜏 )
(we take ℎ(∞) = ∞) is a stopping time.
Indeed,
{𝜎 ≤ 𝑡} = {𝜏 ≤ 𝑡∗ },
(6.5.17)
(6.5.18)
where 𝑡∗ is the largest value of 𝑠 ≤ 𝑡 for which ℎ(𝑠) ≤ 𝑡 if there are such values:
𝑡∗ = max{𝑠 ≤ 𝑡 : ℎ(𝑠) ≤ 𝑡};
(6.5.19)
the maximum is reached because of our restrictions imposed on the function ℎ(𝑠): the set
{𝑠 ≤ 𝑡 : ℎ(𝑠) ≤ 𝑡} is an interval with its right end. If there are no values 𝑠 ≤ 𝑡 for which
ℎ(𝑠) ≤ 𝑡, the event {𝜎 ≤ 𝑡} is just impossible (equal to ∅).
This is a generalization of Example 6.5.5, not of Example 6.5.5′ .
Theorem 6.5.1. If 𝜏 , 𝜎 are stopping times, then so are min(𝜏, 𝜎) and max(𝜏, 𝜎).
Indeed,
{min(𝜏, 𝜎) ≤ 𝑡} = {𝜏 ≤ 𝑡} ∪ {𝜎 ≤ 𝑡},
{max(𝜏, 𝜎) ≤ 𝑡} = {𝜏 ≤ 𝑡} ∩ {𝜎 ≤ 𝑡}. (6.5.20)
6.5.1 For the stochastic process 𝑌𝑡 , 𝑡 ≥ 0, of Example 5.2.1, prove that 𝑇 is a stopping
time with respect to this process.
Check that for a non-random 𝑡∗ > 0 also min(𝑇, 𝑡∗ ) is a stopping time.
4