2072 CHAPTER 62. STOCHASTIC PROCESSES

Theorem 62.5.3 Let X (t) for t ∈ I = [0,T ] be an E valued right continuous martingalewith respect to a filtration, Ft . Then for p≥ 1,

P([

supt∈I∥X (t)∥ ≥ λ

])≤ 1

λp E (∥X (T )∥p) . (62.5.21)

If p > 1,

E

((sup

t∈[S,T ]∥X (t)∥

)p)≤(

pp−1

)E (∥X (T )∥p)

1/p E

((sup

t∈[S,T ]∥X (t)∥

)p)1/p′

(62.5.22)

Proof: By Proposition 62.4.3 ∥X (t)∥ , t ∈ I is a submartingale and so from Theorem62.5.2, it follows 62.5.21 and 62.5.22 hold.

Definition 62.5.4 Let K be a set of functions of L1 (Ω,F ,P). Then K is called equi inte-grable if

limλ→∞

supf∈K

∫[| f |≥λ ]

| f |dP = 0.

Recall that from Corollary 20.9.6 on Page 640 such an equi integrable set of functionsis weakly sequentially precompact in L1 (Ω,F ,P) in the sense that if { fn}⊆K, there existsa subsequence,

{fnk

}and a function, f ∈ L1 (Ω,F ,P) such that for all g ∈ L1 (Ω,F ,P)′ ,

g(

fnk

)→ g( f ) .

62.6 Optional Sampling Theorems62.6.1 Stopping Times And Their PropertiesThe optional sampling theorem is very useful in the study of martingales and submartin-gales as will be shown.

First it is necessary to define the notion of a stopping time.

Definition 62.6.1 Let (Ω,F ,P) be a probability space and let {Fn}∞

n=1 be an increasingsequence of σ algebras each contained in F , called a discrete filtration. A stopping timeis a measurable function, τ which maps Ω to N,

τ−1 (A) ∈F for all A ∈P (N) ,

such that for all n ∈ N,[τ ≤ n] ∈Fn.

Note this is equivalent to saying[τ = n] ∈Fn

because[τ = n] = [τ ≤ n]\ [τ ≤ n−1] .