2058 CHAPTER 62. STOCHASTIC PROCESSES

Then by the Borel Cantelli lemma, there is a set of measure zero N such that for ω /∈ N,

∥X (sn)−X (t)∥α ≤ 2−n

for all n large enough. Then

∥Y (t)−X (t)∥ ≤ ∥Y (t)−Y (sn)∥+∥X (sn)−X (t)∥

which shows that, by continuity of Y, for ω not in an exceptional set of measure zero,∥Y (t)−X (t)∥= 0.

It remains to verify the assertion about Holder continuity of Y . Let 0≤ s < t ≤ T. Thenfor some n,

2−(n+1)T ≤ t− s≤ 2−nT (62.2.14)

Thus∥Y (t)−Y (s)∥ ≤ ∥Y (t)−Xn (t)∥+∥Xn (t)−Xn (s)∥+∥Xn (s)−Y (s)∥

≤ 2 supτ∈[0,T ]

∥Y (τ)−Xn (τ)∥+∥Xn (t)−Xn (s)∥ (62.2.15)

Now∥Xn (t)−Xn (s)∥

t− s≤ ∥Xn (t)−Xn (s)∥

2−(n+1)TFrom 62.2.14 a picture like the following must hold.

s t tn+1k+1tn+1

ktn+1k−1

Therefore, from the above,

∥Xn (t)−Xn (s)∥t− s

≤∥∥X(tn+1k−1

)−X

(tn+1k

)∥∥+∥∥X(tn+1k+1

)−X

(tn+1k

)∥∥2−(n+1)T

≤ C2nMn+1

It follows from 62.2.15,

∥Y (t)−Y (s)∥ ≤ 2∥Y −Xn∥∞+C2nMn+1 (t− s)

Next, letting γ < β/α, and using 62.2.14,

∥Y (t)−Y (s)∥(t− s)γ ≤ 2

(T−12n+1)γ ∥Y −Xn∥∞

+C2n (2−n)1−γ Mn+1

= C2nγ (∥Y −Xn∥∞+Mn+1)

The above holds for any s, t satisfying 62.2.14. Then

sup{∥Y (t)−Y (s)∥

(t− s)γ ,0≤ s < t ≤ T, |t− s| ∈[2−(n+1)T,2−nT

]}