62.7. DOOB OPTIONAL SAMPLING CONTINUOUS CASE 2089

Thus, taking A ∈Fσ and recalling σ ≤ σn so that by Proposition 62.7.8, Fσ ⊆Fσn ,∫A

M (σn∧ τn)dP =∫

AE (M (τn) |Fσn)dP =

∫A

M (τn)dP.

Now passing to a limit as n→∞, the Vitali convergence theorem, Theorem 11.5.3 on Page257 and the right continuity of M implies one can pass to the limit in the above and conclude∫

AM (σ ∧ τ)dP =

∫A

M (τ)dP.

By Proposition 62.7.8, M (σ ∧ τ) is Fσ∧τ ⊆Fσ measurable showing

E (M (τ) |Fσ ) = M (σ ∧ τ) .

A similar theorem is available for submartingales defined on [0,L] ,L≤ ∞.

Theorem 62.7.15 Let {X (t)} be a right continuous submartingale with respect to the in-creasing sequence of σ algebras, {Ft} which is assumed to be a normal filtration,

Ft = ∩s>tFs,

for t ∈ [0,L] , L≤∞ and let σ ,τ be two stopping times with τ bounded. Then X (τ) definedas

ω → X (τ (ω))

is integrable andX (σ ∧ τ)≤ E (X (τ) |Fσ ) .

Proof: Let

τn (ω)≡ ∑k≥0

2−n (k+1)TXτ−1((k2−nT,(k+1)T 2−n]) (ω) .

Then by Lemma 62.7.13 τn is a stopping time, the functions |X (τn)| are uniformly in-tegrable, and |X (τ)| is also integrable. For σn defined similarly to τn, it also follows|X (τn∧σn)| are uniformly integrable.

Let A∈Fσ . Since σ ≤σn, it follows that Fσ ⊆Fσn . By the discrete optional samplingtheorem for submartingales, Theorem 62.6.7,

X (σn∧ τn)≤ E (X (τn) |Fσn)

and so ∫A

X (σn∧ τn)dP≤∫

AE (X (τn) |Fσn)dP =

∫A

X (τn)dP

and now taking limn→∞ of both sides and using the Vitali convergence theorem along withthe right continuity of X , it follows∫

AX (σ ∧ τ)dP≤

∫A

X (τ)dP≡∫

AE (X (τ) |Fσ )dP