2190 CHAPTER 64. WIENER PROCESSES

Lemma 64.4.1 Let (W (t) ,Ft) be a stochastic process which has independent incrementshaving values in E a real separable Banach space. Let

A ∈Fs ≡ σ (W (u)−W (r) : 0≤ r < u≤ s)

Suppose g(W (t)−W (s)) ∈ L1 (Ω;E) . Then the following formula holds.∫Ω

XAg(W (t)−W (s))dP = P(A)∫

Ω

g(W (t)−W (s))dP (64.4.12)

Proof: Let G denote the set, of all A∈Fs such that 64.4.12 holds. Then it is obvious Gis closed with respect to complements and countable disjoint unions. Let K denote thosesets which are finite intersections of the form

A = ∩mi=1Ai

where each Ai is in a set of σ (W (ui)−W (ri)) for some 0 ≤ ri < ui ≤ s. For such A, itfollows

A ∈ σ (W (ui)−W (ri) , i = 1, · · · ,m) .

Now consider the random vector having values in Em+1,

(W (u1)−W (r1) , · · · ,W (um)−W (rm) ,g(W (t)−W (s)))

Let t∗ ∈ (E ′)m and s∗ ∈ E ′.

t∗ · (W (u1)−W (r1) , · · · ,W (um)−W (rm))

can be written in the form g∗ · (W (τ1)−W (η1) , · · · ,W (τ l)−W (η l)) where the intervals,(η j,τ j

)are disjoint and each τ j ≤ s. For example, suppose you have

a(W (2)−W (1))+b(W (2)−W (0))+ c(W (3)−W (1)) ,

where obviously the increments are not disjoint. Then you would write the above expres-sion as

a(W (2)−W (1))+b(W (2)−W (1))+b(W (1)−W (0))+c(W (3)−W (2))+ c(W (2)−W (1))

and then you would collect the terms to obtain

b(W (1)−W (0))+(a+b+ c)(W (2)−W (1))+ c(W (3)−W (2))

and now these increments are disjoint.Therefore, by independence of the increments,

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

= E (exp i(g∗ · (W (τ1)−W (η1) , · · · ,W (τ l)−W (η l))+ s∗ (g(W (t)−W (s)))))