846 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

Theorem 31.4.3 Let {X (t)} be a right continuous nonnegative sub-martingale ad-apted to the normal filtration Ft for t ∈ [0,T ]. Let p≥ 1. Define

X∗ (t)≡ sup{X (s) : 0 < s < t} , X∗ (0)≡ 0.

Then for λ > 0, if X (t)p is in L1 (Ω) for each t,

P([X∗ (T )> λ ])≤ 1λ

p

∫X[X∗(T )>λ ]X (T )p dP (31.2)

If X (t) is continuous, the above inequality holds without this assumption. In case p > 1,and X (t) continuous, then for each t ≤ T,(∫

Ω

|X∗ (t)|p dP)1/p

≤ pp−1

(∫Ω

X (T )p dP)1/p

(31.3)

Proof: The first inequality follows from Theorem 30.5.2. However, it can also beobtained a different way using stopping times. First note that from right continuity, X∗ (t) =sup{X (d) : d ∈ D} where D is a dense countable set in (0, t). Therefore, X∗ (t) is alwaysmeasurable.

First I will assume X (t) is a bounded sub-martingale. These certainly exist. Just take abounded stopping time τ and consider Xτ .

Define the stopping time

τ ≡ inf{t > 0 : X (t)> λ}∧T.

(The infimum over an empty set will equal ∞.)This is a stopping time by 31.3.9 because itis just a continuous function of the first hitting time of an open set. Also from the definitionof X∗ in which the supremum is taken over an open interval,

[τ < t] = [X∗ (t)> λ ]

Note this also shows X∗ (t) is Ft measurable. Then it follows that X p (t) is also a sub-martingale since rp is increasing and convex. By the optional sampling theorem,

X (0)p ,X (τ)p ,X (T )p

is a sub-martingale. Recall X (σ ∧ τ)≤ E (X (τ) |Fσ ) when τ is bounded. I need to verifythat

E (X (T )p |Fτ)≥ X (τ)p ,E (X (τ)p |F0)≥ X (0)p .

But from the optional sampling theorem Theorem 31.3.16

E (X (T )p |Fτ) ≥ X (T ∧ τ)p = X (τ)p

E (X (τ)p |F0) ≥ X (τ ∧0)p = X (0)p

Also [τ < T ] ∈Fτ . Recall that A is Fτ measurable means A∩ [τ ≤ t] ∈Ft . Since τ isa stopping time, [τ ≤ T ]∩ [τ ≤ t] = [τ ≤ t] ∈Ft and so [τ ≤ T ] ∈Fτ .∫

[τ<T ]X (τ)p dP≤

∫[τ<T ]

E (X (T )p |Fτ)dP =∫[τ<T ]

X (T )p dP