826 CHAPTER 30. CONTINUOUS STOCHASTIC PROCESSES

2. If g is an increasing convex function from [0,∞) to [0,∞) and

E (g(∥X (t)∥))< ∞

for all t ∈ [0,T ] then g(∥X (t)∥) , t ∈ [0,T ] is a sub-martingale.

Proof: Let s≤ t. Then from properties of conditional expecation and Theorem 24.12.1on Page 702,

∥X (s)∥ = ∥E (X (s)−X (t) |Fs)+E (X (t) |Fs)∥

≤=0 a.e.︷ ︸︸ ︷

∥E (X (s)−X (t) |Fs)∥+∥E (X (t) |Fs)∥ ≤ ∥E (X (t) |Fs)∥≤ E (∥X (t)∥|Fs)

Consider the second claim. Recall Jensen’s inequality for sub-martingales, Theorem29.1.7 on Page 784. From the first part

∥X (s)∥ ≤ E (∥X (t)∥|Fs) a.e.

and so from Jensen’s inequality,

g(∥X (s)∥)≤ g(E (∥X (t)∥|Fs))≤ E (g(∥X (t)∥) |Fs) a.e.,

showing that g(∥X (t)∥) is also a sub-martingale. ■

30.5 Some Maximal EstimatesMartingales and sub-martingales have some very interesting maximal estimates. I willpresent some of these here. The proofs are fairly general. For convenience, assume each Ftcontains the sets of measure zero from F . This is so that it suffices to assume t→ X (t)(ω)is right continuous off some set of measure zero. If it were right continuous for each ω,then it wouldn’t matter. Actually, in this book, I will mainly be interested in continuousprocesses. It is also possible to show that for real valued processes, one can get a rightcontinuous version but this will not be used.

Lemma 30.5.1 Let {Ft} be a filtration and let {X (t)} be a nonnegative valued sub-martingale for t ∈ [S,T ] . Then for λ > 0 and any p≥ 1, if, for each t, At is a Ft measurablesubset of [X (t)> λ ] , then

P(At)≤1

λp

∫At

X (T )p dP.

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 ■

The following theorem is the main result.