2084 CHAPTER 62. STOCHASTIC PROCESSES

Thus A ∈Fτ and so Fσ ⊆Fτ .Consider the following approximation of τ in which tn

k = k2−n.

τn ≡∞

∑k=1

X[τ∈(tnk ,t

nk+1]]

tnk+1

Thus τn ↓ τ . Consider for U an open set, X (τn)−1 (U)∩ [τn < t] . Say t ∈ (tn

k , tnk+1]. Then

from the above definition of τn,

[τn < t] = [τ ≤ tnk ] ∈Ftn

k⊆Ft

It follows that

X (τn)−1 (U)∩ [τn < t] = ∪k

j=1

∈Ftnj

X(tn

j)−1

(U)∩∈Ftnj[

τn = tnj]

and so this set is in Ftnk⊆Ft . The reason[

τn = tnj]∈Ftn

j

is that it equals[τ ∈ (tn

j−1, tnj ]]∈Ftn

jby assumption that τ is a stopping time.

By right continuity of X , it follows that

X (τ)−1 (U)∩ [τ < t] = ∪∞m=1∩n≥m X (τn)

−1 (U)∩ [τn < t] ∈Ft

It follows that for every m,

X (τ)−1 (U)∩ [τ ≤ t] = ∩∞n=mX (τ)−1 (U)∩

[τ < t +

1n

]∈Ft+ 1

m

Since the filtration is normal, it follows that

X (τ)−1 (U)∩ [τ ≤ t] ∈Ft+ = Ft .

Now consider an increasing family of stopping times, τ (t) (ω→ τ (t)(ω)). It turns outthis is a submartingale.

Example 62.7.9 Let {τ (t)} be an increasing family of stopping times. Then τ (t) is adap-ted to the σ algebras Fτ(t) and {τ (t)} is a submartingale adapted to these σ algebras.

First I need to show that a stopping time, τ is Fτ measurable. Consider [τ ≤ s] . Isthis in Fτ ? Is [τ ≤ s]∩ [τ ≤ r] ∈Fr for each r? This is obviously so if s ≤ r because theintersection reduces to [τ ≤ s] ∈Fs ⊆Fr. On the other hand, if s > r then the intersectionreduces to [τ ≤ r] ∈Fr and so it is clear that τ is Fτ measurable. It remains to verify it isa submartingale.

Let s < t and let A ∈Fτ(s)∫A

E(τ (t) |Fτ(s)

)dP≡

∫A

τ (t)dP≥∫

Aτ (s)dP