2080 CHAPTER 62. STOCHASTIC PROCESSES
62.7 Doob Optional Sampling Continuous Case62.7.1 Stopping TimesLet X (t) be a stochastic process adapted to a filtration {Ft} for t ∈ [0,T ]. We will assumetwo things. The stochastic process is right continuous and the filtration is normal.
Definition 62.7.1 A normal filtration is one which satisfies the following :
1. F0 contains all A ∈F such that P(A) = 0. Here F is the σ algebra which containsall Ft .
2. Ft = Ft+ for all t ∈ I where Ft+ ≡ ∩s>tFs.
For an F measurable [0,∞) valued function τ to be a stopping time, we want to havethe stopped process Xτ defined by Xτ (t)(ω)≡ X (t ∧ τ (ω))(ω) to be adapted whenever Xis right continuous and adapted. Thus a stopping time is a measurable function which canbe used to stop the process while retaining the property of being adapted. We want to finda simple criterion which will ensure that this happens.
Let X (t) be adapted. Let O be an open set in some metric space where X has its values.This could probably be generalized. Then we need to have
Xτ (t)−1 (O) ∈Ft
Thus we need to have
[τ ≤ t]∩[X (τ)−1 (O)
]∪ [τ > t]∩
[X (t)−1 (O)
]∈Ft
How does this happen? Consider τk (ω)≡∑∞n=0 X
τ−1((n2−k,(n+1)2−k]) (ω)(n+1)2−k. Thusfor each ω,τk (ω) ↓ τ (ω). Since X is right continuous for each ω, it follows that, since Ois open,
[τ ≤ t]∩[X (τ)−1 (O)
]= [τ ≤ t]∩
(∪m∩k≥m
[X (τk)
−1 (O)])
= ∪m∩k≥m∪∞n=0τ
−1((n2−k,(n+1)2−k ∧ t]
)∩[
X((n+1)2−k
)−1(O)
]the last union being a disjoint union. Now
τ−1((n2−k,(n+1)2−k ∧ t]
)∩[
X((n+1)2−k
)−1(O)
]is a set of F(n+1)2−k intersected with
[τ ∈ (n2−k,(n+1)2−k]
]∩ [τ ∈ (0, t]] . If we assume
[τ ≤ t] ∈Ft , for all t, then this shows that the above expression is a set of Ft+2−k . Since
this is true for each k, and the filtration is normal, this implies [τ ≤ t]∩[X (τ)−1 (O)
]∈Ft .
Also with this assumption, [τ > t] = [τ ≤ t]C ∈Ft and so we get Xτ (t)−1 (O) ∈Ft . Thisis why we define stopping times this way. It is so that when you have a right continuousadapted process, then the stopped process is also adapted.