With continuous processes, the discrete filtration is replaced by a normal filtration. Also
we tend to feature right continuous or continuous processes. As in the case of discrete
martingales, there is something called a stopping time.
Definition 61.7.1Let
(Ω,ℱ,P )
be a probability space and let ℱ_{t}be a filtration. Ameasurable function, τ : Ω →
[0,∞ ]
is called a stopping time if
[τ ≤ t] ∈ ℱ
t
for all t ≥ 0. Associated with a stopping time is the σ algebra, ℱ_{τ}defined by
ℱ ≡ {A ∈ ℱ : A∩ [τ ≤ t] ∈ ℱ for all t}.
τ t
These sets are also called those “prior” to τ.
Note that ℱ_{τ} is obviously closed with respect to countable unions. If A ∈ℱ_{τ},
then
AC ∩ [τ ≤ t] = [τ ≤ t]∖ (A ∩ [τ ≤ t]) ∈ ℱ
t
Thus ℱ_{τ} is a σ algebra.
Proposition 61.7.2Let B be an open subset of topological space E and let X
(t)
be aright continuous ℱ_{t}adapted stochastic process such that ℱ_{t}is normal. Thendefine
τ (ω) ≡ inf{t > 0 : X (t)(ω) ∈ B }.
This is called the first hitting time.Then τ is a stopping time. If X
(t)
is continuous andadapted to ℱ_{t}, a normal filtration, then if H is a nonempty closed set such thatH = ∩_{n=1}^{∞}B_{n}for B_{n}open, B_{n}⊇ B_{n+1},
τ (ω) ≡ inf{t > 0 : X (t)(ω) ∈ H}
is also a stopping time.
Proof: Consider the first claim. ω ∈
[τ = a]
implies that for each n ∈ ℕ, there exists
t ∈
[a,a + 1]
n
such that X
(t)
∈ B. Also for t < a, you would need X
(t)
∕∈
B. By right
continuity, this is the same as saying that X
(d)
∕∈
B for all rational d < a. (If t < a, you
could let d_{n}↓ t where X
(d )
n
∈ B^{C}, a closed set. Then it follows that X
(t)
is
also in the closed set B^{C}.) Thus, aside from a set of measure zero, for each
m ∈ ℕ,
( )
∩∞n=m ∪d∈ℚ∩ a,a+1 [X (d) ∈ B] ∩(∩d∈ℚ∩[0,a)[X (d) ∈ BC ]) ∈ ℱa+ 1
[ n] m
Thus, since the filtration is normal,
[τ = a] ∈ ∩∞m=1 ℱa+1-= ℱa+ = ℱa
m
Now what of
[τ < a]
? This is equivalent to saying that X
(t)
∈ B for some t < a. Since X
is right continuous, this is the same as saying that X
(t)
∈ B for some t ∈ ℚ,t < a.
Thus
[τ < a] = ∪ [X (d) ∈ B] ∈ ℱ
d∈ℚ,d<a a
It follows that
[τ ≤ a]
=
[τ < a]
∪
[τ = a]
∈ℱ_{a}.
Now consider the claim involving the additional assumption that X
(t)
is continuous
and it is desired to hit a closed set H = ∩_{n=1}^{∞}B_{n} where B_{n} is open, B_{n}⊇ B_{n+1}. (Note
that if the topological space is a metric space, this is always possible so this is not a big
restriction.) Then let τ_{n} be the first hitting time of B_{n} by X
(t)
. Then it can be shown
that
[τ ≤ a] = ∩n [τn ≤ a] ∈ ℱa
To show this, first note that ω ∈
[τ ≤ a]
if and only if there exists t ≤ a such that
X
(t)
(ω)
∈ H. This follows from continuity and the fact that H is closed. Thus τ_{n}≤ a
for all n because for some t ≤ a, X
(t)
∈ H ⊆ B_{n} for all n. Next suppose ω ∈
[τn ≤ a]
for
all n. Then for δ_{n}↓ 0, there exists t_{n}∈
[0,a+ δn]
such that X
(tn)
(ω)
∈ B_{n}. It follows
there is a subsequence, still denoted by t_{n} such that t_{n}→ t ∈
[0,a]
. By continuity of X, it
must be the case that X
(t)
(ω)
∈ H and so ω ∈
[τ ≤ a]
. This shows the above
formula. Now by the first part, each
[τn ≤ a]
∈ℱ_{a} and so
[τ ≤ a]
∈ℱ_{a} also.
■
Another useful result for real valued stochastic process is the following.
Proposition 61.7.3Let X
(t)
be a real valued stochastic process which is ℱ_{t}adaptedfor a normal filtration ℱ_{t}, with the property that off a set of measure zero in Ω, t → X
∈ℱ_{a}, the equality following from lower
semicontinuity. Thus
[τ ≤ a]
=
[τ = a]
∪
[τ < a]
∈ℱ_{a}. ■
Thus there do exist stopping times, the first hitting time above being an example.
When dealing with continuous stopping times on a normal filtration, one uses the
following discrete stopping times
∞∑ n
τn ≡ X[τ∈(tnk,tnk+1]]tk+1
k=1
where here
|| n n ||
tk − tk+1
= r_{n} for all k where r_{n}→ 0. Then here is an important
lemma.
Lemma 61.7.4τ_{n}is a stopping time (
[τn ≤ t]
∈ ℱ_{t}.) Also ℱ_{τ}⊆ ℱ_{τn}and foreach ω,τ_{n}
(ω)
↓ τ
(ω)
.
Proof:Say t ∈ (t_{k−1}^{n},t_{k}^{n}]. Then
[τn ≤ t]
=
[ ]
τ ≤ tnk−1
if t < t_{k}^{n} and it
equals
[τ ≤ tnk]
if t = t_{k}^{n}. Either way
[τn ≤ t]
∈ℱ_{t} so it is a stopping time. Also
from the definition, it follows that τ_{n}≥ τ and
|τn (ω)− τ (ω)|
≤ r_{n} which is
given to converge to 0. Now suppose A ∈ℱ_{τ} and say t ∈ (t_{k−1}^{n},t_{k}^{n}] as above.
Then
[ ]
A ∩[τn ≤ t] = A ∩ τ ≤ tnk−1 ∈ ℱtnk−1 ⊆ ℱt if t < tnk
and
A∩ [τn ≤ t] = A ∩[τ ≤ tnk] ∈ ℱtn= ℱt if t = tnk
k
Thus ℱ_{τ}⊆ℱ_{τn} as claimed. ■
Next is the claim that if X
(t)
is adapted to ℱ_{t}, then X
(τ )
is adapted to ℱ_{τ} just like
the discrete case.
Proposition 61.7.5Let
(Ω,ℱ,P )
be a probability space and let σ ≤ τ be twostopping times with respect to a filtration, ℱ_{t}. Then ℱ_{σ}⊆ℱ_{τ}. If X
(t)
is a rightcontinuous stochastic process adapted to a normal filtration ℱ_{t}and τ is a stoppingtime, ω → X