64.4. INDEPENDENT INCREMENTS AND MARTINGALES 2191

=l

∏j=1

E(exp(ig j(W (τ j)−W

(η j))))

E (exp(is∗ (g(W (t)−W (s)))))

= E (exp(i(t∗ · (W (u1)−W (r1) , · · · ,W (um)−W (rm))))) ·E (exp(is∗ (g(W (t)−W (s))))) .

By Theorem 59.13.3, it follows the vector (W (u1)−W (r1) , · · · ,W (um)−W (rm)) is inde-pendent of the random variable g(W (t)−W (s)) which shows that for A ∈K , XA, mea-surable in σ (W (u1)−W (r1) , · · · ,W (um)−W (rm)) is independent of g(W (t)−W (s)) .Therefore, ∫

Ω

XAg(W (t)−W (s))dP =∫

Ω

XAdP∫

Ω

g(W (t)−W (s))dP

= P(A)∫

Ω

g(W (t)−W (s))dP

Thus K ⊆ G and so by the lemma on π systems, Lemma 12.12.3 on Page 329, it followsG ⊇ σ (K )⊇Fs ⊇ G .

Lemma 64.4.2 Let {W (t)} be a stochastic process having values in a separable Banachspace which has the property that if t1 < t2 · · ·< tn, then the increments,

{W (tk)−W (tk−1)}

are independent and integrable and E (W (t)−W (s)) = 0. Suppose also that W (t) is rightcontinuous, meaning that for ω off a set of measure zero, t→W (t)(ω) is right continuous.Also suppose that for some q > 1

||W (t)−W (s)||Lq(Ω)

is bounded independent of s ≤ t. Then {W (t)} is also a martingale with respect to thenormal filtration defined by

Fs ≡ ∩t>sσ (W (u)−W (r) : 0≤ r < u≤ t)

where this denotes the intersection of the completions of the σ algebras

σ (W (u)−W (r) : 0≤ r < u≤ t)

Also, in the same situation but without the assumption that E (W (t)−W (s)) = 0, if t > sand A ∈Fs it follows that if g is a continuous function such that

||g(W (t)−W (s))||Lq(Ω) (64.4.13)

is bounded independent of s≤ t for some q > 1 then for t > s,∫Ω

XAg(W (t)−W (s))dP = P(A)∫

Ω

g(W (t)−W (s))dP. (64.4.14)