856 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

Right side of 31.13

From 31.14, used to substitute for P([τa < τb]) this yields

0≤ bP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa])

= bP([τa = τb]∩ [M∗ > 0])+a [P([M∗ > 0])−P([τa ≥ τb]∩ [M∗ > 0])]+bP([τb < τa])

= bP([τa ≥ τb]∩ [M∗ > 0])+a [P([M∗ > 0])−P([τa ≥ τb]∩ [M∗ > 0])]

and so(b−a)P([τa ≥ τb])≥−aP([M∗ > 0]) (31.17)

Next use 31.14 to substitute for the term P([τb < τa]) and write

0≤ bP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])+bP([τb < τa])

= bP([τa = τb]∩ [M∗ > 0])+aP([τa < τb])

+b [P([M∗ > 0])−P([τa ≤ τb]∩ [M∗ > 0])]

= aP([τa < τb])+bP([M∗ > 0])−bP([τa < τb]∩ [M∗ > 0])= aP([τa < τb])+bP([M∗ > 0])−bP([τa < τb])

and so(b−a)P([τa < τb])≤ bP([M∗ > 0]) (31.18)

Now the boxed in formulas in 31.15 - 31.18 yield the conclusion of the theorem. ■Note P([τa < τb]) means M (t) hits a before it hits b with other occurrences of similar

expressions being defined similarly.

31.7 The Space M pT (E)

Here p≥ 1. Also, we assume the filtration is a normal filtration.

Definition 31.7.1 Then M ∈M pT (E) if t→M (t)(ω) is continuous for a.e. ω and

M (t) is adapted, and

E

(sup

t∈[0,T ]∥M (t)∥p

)< ∞

Here E is a separable Banach space.

Proposition 31.7.2 Define a norm on M pT (E) by

∥M∥M pT (E) ≡ E

(sup

t∈[0,T ]∥M (t)∥p

)1/p

.

Then with this norm, M pT (E) is a Banach space. Also, a Cauchy sequence in this space has

a subsequence which converges uniformly for all ω off a set of measure zero. Those M inM p

T (E) which are martingales constitute a closed subspace of M pT (E). If σ is a stopping

time, then if M ∈M pT (E), so is Mσ and ∥M∥M p

T (E) ≥ ∥Mσ∥M pT (E). Thus if Mn → M in

M pT (E) , then Mσ

n →Mσ in M pT (E) .