2070 CHAPTER 62. STOCHASTIC PROCESSES
Proof: From Jensen’s inequality,
λpP(At) ≤
∫At
X (t)p dP≤∫
At
E (X (T ) |Ft)p dP
≤∫
At
E (X (T )p |Ft)dP =∫
At
X (T )p dP
and this proves the lemma.The following theorem is the main result.
Theorem 62.5.2 Let {Ft} be a filtration and let {X (t)} be a nonnegative valued rightcontinuous1 submartingale 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∗
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 62.5.1,
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
≤ 1λ
p
∫Ω
X[supt∈D X(t)≥λ ]X (T )p dP.
1t→M (t)(ω) is continuous from the right for a.e. ω .