2192 CHAPTER 64. WIENER PROCESSES

Proof: Consider first the claim, 64.4.14. To begin with I show that if A ∈Fs then forall ε small enough that t > s+ ε,1∫

Ω

XAg(W (t)−W (s+ ε))dP = P(A)∫

Ω

g(W (t)−W (s+ ε))dP (64.4.15)

This will happen if XA and g(W (t)−W (s+ ε)) are independent. First note that from thedefinition

A ∈ σ (W (u)−W (r) : 0≤ r < u≤ s+ ε)

and so from the process of completion of a measure space, there exists

B ∈ σ (W (u)−W (r) : 0≤ r < u≤ s+ ε)

such that B⊇ A and P(B\A) = 0. Therefore, letting φ ∈ E ′,

E (exp(itXA + iφ (g(W (t)−W (s+ ε)))))

= E (exp(itXB + iφ (g(W (t)−W (s+ ε)))))

= E (exp(itXB))E (exp(iφ (g(W (t)−W (s+ ε)))))

because XB is independent of g(W (t)−W (s+ ε)) by Lemma 64.4.1 above. Then theabove equals

= E (exp(itXA))E (exp(iφ (g(W (t)−W (s+ ε)))))

Now by Theorem 59.13.3, 64.4.15 follows. Next pass to the limit in both sides of 64.4.15as ε→ 0. One can do this because of 64.4.13 which implies the functions in the integrandsare uniformly integrable and Vitali’s convergence theorem, Theorem 21.5.7. This yields64.4.14.

Now consider the part about the stochastic process being a martingale. Let g be theidentity map. If A ∈Fs, the above implies∫

AE (W (t) |Fs)dP =

∫A

W (t)dP =∫

A(W (t)−W (s))dP+

∫A

W (s)dP

= P(A)∫

Ω

(W (t)−W (s))dP+∫

AW (s)dP =

∫A

W (s)dP

and so since A is arbitrary, E (W (t) |Fs) =W (s).Note this implies immediately from Lemma 63.1.5 that Wiener process is not of boun-

ded variation on any interval. This is because this lemma implies if it were of boundedvariation, then it would be constant which is not the case due to

L (W (t)−W (s)) = L (W (t− s)) = L(√

t− sW (1)).

Here is an interesting theorem about approximation.

1Note how the σ algebra Fs are defined, as the intersection of completions of σ algebras corresponding to tstrictly larger than s.