2252 CHAPTER 65. STOCHASTIC INTEGRATION

The elementary functions {Φn} have values in L (U1,H)0 meaning they are restrictions offunctions in L (U1,H) to JQ1/2U and converge to Φ◦ J−1 in

L2([a,T ]×Ω;L2

(JQ1/2U,H

))where Φ ∈ L2

([a,T ]×Ω;L2

(Q1/2U,H

))is given. Let

Φ(t)≡n−1

∑k=0

Φ(tk)X(tk,tk+1] (t)

be an elementary function. In particular, let Φ(tk) be Ftk measurable as a map intoL (U1,H), and has finitely many values. As just mentioned, the topic of interest is theelementary functions Φn in the above diagram. Thus Φ will be one of these elementaryfunctions.

Let τ be a stopping time having values from the set of mesh points {tk} for the elemen-tary function. Then from the definition of the integral for elementary functions,

∫ t∧τ

aΦdW ≡

n−1

∑k=0

Φ(tk)(W (t ∧ τ ∧ tk+1)−W (t ∧ τ ∧ tk))

If ω is such that τ (ω) = t j, then to get something nonzero, you must have t j > tk sok ≤ j−1. Thus the above on the right reduces to

j−1

∑k=0

Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk))

It clearly is 0 if j = 0. Define ∑−1k=0 ≡ 0. Thus the integral equals

n

∑j=0

X[τ=t j]

j−1

∑k=0

Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk))

Interchanging the order of summation, k ≤ j−1 so j ≥ k+1 and this equals

n−1

∑k=0

n

∑j=k+1

X[τ=t j]Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk))

=n−1

∑k=0

Ftk measurable︷ ︸︸ ︷X[τ>tk]Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk))

Therefore ∫ t∧τ

aΦdW =

∫ t

a

n−1

∑k=0

X[τ>tk]Φ(tk)X(tk,tk+1]dW (65.8.15)