2088 CHAPTER 62. STOCHASTIC PROCESSES
Because P(A(τn)> λ )→ 0 and a single function in L1 is uniformly integrable. Thusthese functions A(τn) are equi-integrable. Hence they are uniformly integrable. Now tn
k →∣∣M (tnk
)∣∣ is also a nonnegative submartingale. Thus
|M (0)| , |M (τn)| , |M (T )|
is a submartingale by the optional sampling theorem for discrete submartingales given ear-lier. Therefore,
P(|M (τn)|> λ )≤ 1λ
∫[|M(τn)|>λ ]
|M (τn)|dP≤ 1λ
∫[|M(τn)|>λ ]
|M (T )|dP≤ ∥M (T )∥L1
λ
Of course ∥M (T )∥L1 is finite because it is dominated by∫Ω
A(T )+ |X (T )|dP < ∞
Hence
limλ→∞
supn
∫[|M(τn)|>λ ]
|M (τn)|dP≤ limλ→∞
supn
∫[|M(τn)|>λ ]
|M (T )|dP = 0
because a single function in L1 is uniformly integrable and the above estimate shows thatP([|M (τn)|> λ ])→ 0 uniformly in n. Thus, in fact X (τn) must be uniformly integrablesince it is the sum of two which are.
Theorem 62.7.14 Let {M (t)} be a right continuous martingale having values in E a sepa-rable real Banach space with respect to the increasing sequence of σ algebras, {Ft} whichis assumed to be a normal filtration satisfying,
Ft = ∩s>tFs,
for t ∈ [0,L] , L≤∞ and let σ ,τ be two stopping times with τ bounded. Then M (τ) definedas
ω →M (τ (ω))
is integrable andM (σ ∧ τ) = E (M (τ) |Fσ ) .
Proof: Since M (t) is a martingale, ∥M (t)∥ is a submartingale. Let
τn (ω)≡∞
∑k=0
2−n (k+1)TXτ−1((k2−nT,(k+1)T 2−n]) (ω) .
By Lemma 62.7.13, τn is a stopping time and the functions ||M (τn)|| are uniformly in-tegrable. Also ||M (τ)|| is integrable. Similarly ||M (τn∧σn)|| are uniformly integrablewhere σn is defined similarly to τn.
Consider the main claim now. Letting σ ,τ be stopping times with τ bounded, it followsthat for σn and τn as above, it follows from Theorem 62.6.5
M (σn∧ τn) = E (M (τn) |Fσn)