2098 CHAPTER 62. STOCHASTIC PROCESSES

Proof: Let{

tn0 , t

n1 , · · · , tn

mn

}be a partition of [0,T ] in which

∣∣tni − tn

i−1

∣∣ < ρn whereρn→ 0. Now define Xn as follows:

Xn (t)(ω) ≡mn

∑i=1

X (tni )(ω)X(tn

i−1,tni ](t)

Xn (0) ≡ X (0) .

then each Xn is obviously product measurable because it is the sum of functions which are.By right continuity, Xn converges pointwise to X for ω /∈ N where N is a set of measurezero and so if Y (t)(ω)≡ X (t)(ω) for all ω /∈ N and Y (t)(ω) = 0 for all ω ∈ N, this is thedesired product measurable function.

To see the last claim, let s be a nonnegative simple function, s(ω) = ∑nk=1 ckXEk (ω)

where the ck are strictly increasing in k. Also let Fk = ∪ni=kEi. Then

X[s>λ ] =n

∑k=1

X[ck−1,ck) (λ )XFk (ω)

which is clearly product measurable. For arbitrary f ≥ 0 and measurable, there is an in-creasing sequence of simple functions sn converging pointwise to f . Therefore,

limn→∞

X[sn>λ ] = X[ f>λ ]

and so X[ f>λ ] is product measurable.

Definition 62.9.2 Let X (t) be a right continuous submartingale for t ∈ I and let {τn} bea sequence of stopping times such that limn→∞ τn = ∞. Then Xτn is called the stoppedsubmartingale and it is defined by

Xτn (t)≡ X (t ∧ τn) .

Proposition 62.9.3 The stopped submartingale just defined is a submartingale.

Proof: By the optional sampling theorem for submartingales, Theorem 62.7.15, it fol-lows that for s < t,

E (Xτn (t) |Fs) ≡ E (X (t ∧ τn) |Fs)≥ X (t ∧ τn∧ s)

= X (τn∧ s)≡ Xτn (s) .

Theorem 62.9.4 Let {X (t)} be a right continuous nonnegative submartingale adapted tothe 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 (62.9.31)