2110 CHAPTER 62. STOCHASTIC PROCESSES

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

and so(b−a)P([τa ≥ τb])≥−aP([M∗ > 0]) (62.11.47)

Next use 62.11.44 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]) (62.11.48)

Now the boxed in formulas in 62.11.45 - 62.11.48 yield the conclusion of the theorem. Thisproves the theorem.

Note P([τa < τb]) means M (t) hits a before it hits b with other occurrences of similarexpressions being defined similarly.

62.12 The Space M pT (E)

Here p≥ 1.

Definition 62.12.1 Let M be an E valued martingale. Then M ∈M pT (E) if t→M (t)(ω)

is continuous for a.e. ω and

E

(sup

t∈[0,T ]||M (t)||p

)< ∞

Here E is a separable Banach space.

Proposition 62.12.2 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.

Proof: First it is good to observe that supt∈[0,T ] ||M (t)||p is measurable. This followsbecause of the continuity of t →M (t) . Let D be a dense countable set in [0,T ] . Then bycontinuity,

supt∈[0,T ]

||M (t)||p = supt∈D||M (t)||p