2056 CHAPTER 62. STOCHASTIC PROCESSES

Theorem 62.2.3 Suppose X is a stochastic process on [0,T ] having values in the Banachspace E. Suppose also that there exists a constant, C and positive numbers α,β ,α ≥ 1,such that

E(||X (t)−X (s)||α

)≤C |t− s|1+β (62.2.10)

Then there exists a stochastic process Y such that for a.e. ω, t → Y (t)(ω) is Hölder con-tinuous with exponent γ < β

αand for each t, P([||X (t)−Y (t)||> 0]) = 0. (Y is a version

of X .) Also

E(

sup0≤s<t≤T

∥Y (t)−Y (s)∥(t− s)γ

)≤C

where C depends on α,β ,T,γ .

Proof: The proof considers piecewise linear approximations of X which are automat-ically continuous. These are shown to converge to Y in Lα (Ω;C ([0,T ] ,E)) so it followsthat Y must be continuous for a.e. ω . Finally, it is shown that Y is a version of X andis Holder continuous. In the proof, I will use C to denote a constant which depends onthe quantities γ,α,β ,T . Let

{tnk

}2n

k=0 be a uniform partition of the interval [0,T ] so thattnk+1− tn

k = T 2−n. Now let

Mn ≡maxk≤2n

∥∥X (tnk )−X

(tnk−1)∥∥

Then it follows that

Mαn ≤

2n

∑k=1

∥∥X (tnk )−X

(tnk−1)∥∥α

and so

E (Mαn )≤

2n

∑k=1

C(T 2−n)1+β

=C2n2−n(1+β ) =C2−nβ (62.2.11)

Next denote by Xn the piecewise linear function which results from the values of X atthe points tn

k . Consider the following picture which illustrates a part of the graphs of Xn andXn+1.

tnk−1 tn

ktn+12k−2 tn+1

2ktn+12k−1

Then

maxt∈[0,T ]

∥Xn+1 (t)−Xn (t)∥ ≤ max1≤k≤2n+1

∥∥∥∥∥X(tn+12k−1

)−

X(tnk

)+X

(tnk−1

)2

∥∥∥∥∥