30.5. SOME MAXIMAL ESTIMATES 827
Theorem 30.5.2 Let {Ft} be a filtration and let {X (t)} be a nonnegative valuedright continuous1 sub-martingale for t ∈ [S,T ] . Then for all λ > 0 and p≥ 1, for
X∗ ≡ supt∈[S,T ]
X (t) ,
P([X∗ > λ ])≤ 1λ
p
∫Ω
X[X∗>λ ]X (T )p dP
In the case that p > 1, it is also true that
E ((X∗)p)≤(
pp−1
)E (X (T )p)
1/p(E ((X∗)p))
1/p′
Also there are no measurability issues related to the above supt∈[S,T ] X (t)≡ X∗. If X (t) ∈Lp (Ω) for each t, then
E ((X∗)p)1/p ≤
(p
p−1
)E (X (T )p)
1/p
Thus X∗ is also in Lp (Ω).
Proof: Let S≤ tm0 < tm
1 < · · ·< tmNm
= T where tmj+1− tm
j = (T −S)2−m. First considerm = 1.
At10≡{
ω ∈Ω : X(t10)(ω)> λ
}, At1
1≡{
ω ∈Ω : X(t11)(ω)> λ
}\At1
0
At12≡{
ω ∈Ω : X(t12)(ω)> λ
}\(
At10∪At1
0
).
Do this type of construction for m = 2,3,4, · · · yielding disjoint sets,{
Atmj
}2m
j=0whose
union equals∪t∈Dm [X (t)> λ ]
where Dm ={
tmj
}2m
j=0. Thus Dm ⊆Dm+1. Then also, D≡∪∞
m=1Dm is dense and countable.
From Lemma 30.5.1,
P(∪t∈Dm [X (t)> λ ]) = P
([sup
t∈Dm
X (t)> λ
])=
2m
∑j=0
P(
Atmj
)≤ 1
λp
2m
∑j=0
∫Atmj
X[supt∈Dm X(t)>λ ]X (T )p dP (30.22)
≤ 1λ
p
∫Ω
X[supt∈Dm X(t)>λ ]X (T )p dP≤ 1λ
p
∫Ω
X[supt∈D X(t)>λ ]X (T )p dP.
Let m→ ∞ in the above to obtain
P(∪t∈D [X (t)> λ ]) = P([
supt∈D
X (t)> λ
])≤ 1
λp
∫Ω
X[supt∈D X(t)>λ ]X (T )p dP. (30.23)
1t→ X (t)(ω) is continuous from the right for a.e. ω .