30.3. FILTRATIONS 817

and so there exists a set of measure zero N such that for ω /∈ N,

sup0≤s<t≤T

(∥Y (t)−Y (s)∥

(t− s)γ

)α

≤ 2αk

for all k large enough. Pick such a k, depending on ω /∈ N. Then for any s, t,

∥Y (t)−Y (s)∥(t− s)γ ≤ 2k

and so, this has shown that for ω /∈ N, ∥Y (t)−Y (s)∥ ≤C (ω)(t− s)γ ■Note that if X (t) is known to be continuous off a set of measure zero, then the piece-

wise linear approximations converge to X (t) in C ([0,T ] ,E) off this set of measure zero.Therefore, it must be that off a set of measure zero, Y (t) = X (t) and so in fact X (t) isHolder continuous off a set of measure zero and the condition on expectation also musthold, that is

E(

sup0≤s<t≤T

∥X (t)−X (s)∥(t− s)γ

)≤C.

30.3 FiltrationsInstead of having a sequence of σ algebras, one can consider an increasing collection of σ

algebras indexed by t ∈ R. This is called a filtration.

Definition 30.3.1 Let X be a stochastic process defined on an interval, I = [0,T ]or [0,∞). Suppose the probability space, (Ω,F ,P) has an increasing family of σ algebras,{Ft}. This is called a filtration. If for arbitrary t ∈ I the random variable X (t) is Ft mea-surable, then X is said to be adapted to the filtration {Ft}. Denote by Ft+ the intersectionof all Fs for s > t. The filtration is normal if

1. F0 contains all A ∈F such that P(A) = 0

2. Ft = Ft+ for all t ∈ I.

X is called progressively measurable if for every t ∈ I, the mapping

(s,ω) ∈ [0, t]×Ω, (s,ω)→ X (s,ω)

is B([0, t])×Ft measurable.

Thus X is progressively measurable means

(s,ω)→X[0,t] (s)X (s,ω)

is B([0, t])×Ft measurable. As an example of a normal filtration, here is an example.

Example 30.3.2 For example, you could have a stochastic process, X (t) and you coulddefine

Gt ≡ σ (X (s) : s≤ t),