2266 CHAPTER 65. STOCHASTIC INTEGRATION

By localization, this is ∫ t

0X[0,τn

p]R((

Z ◦ J−1)∗(X ln

))◦ JdW

If ω is not in a suitable set of measure zero, then τnp (ω) ≥ t provided p is large enough.

Thus, for such ω, if p is large enough,

mn−1

∑j=0

(∫ tnj+1∧τn

p∧t

tnj∧τn

p∧tZ (u)dW,X

(tn

j))

H

=∫ t

0X[0,τn

p]R((

Z ◦ J−1)∗(X ln

))◦ JdW

=∫ t

0R((

Z ◦ J−1)∗(X ln

))◦ JdW

This shows that the expression is a local martingale. Also note that the expression on theleft does not depend on J or U1 so the same must be true of the expression on the rightalthough it does not look that way. This has proved the following important theorem.

Theorem 65.14.1 Let Z ∈ L2([0,T ]×Ω,L2

(Q1/2U,H

))and let X ∈ L2 ([0,T ]×Ω,H) ,

both X ,Z progressively measurable. Also let{

tnj

}mn

j=1be a sequence of partitions of [0,T ]

such that each X(

tnj

)is in L2 (Ω,H) . Then

m−1

∑j=0

(∫ tnj+1∧t

tnj∧t

Z (u)dW,X(tn

j))

H

(65.14.21)

is a stochastic integral of the form∫ t

0R((

Z ◦ J−1)∗(X ln

))◦ JdW

where{

τnp}∞

p=1 is a localizing sequence used to define the above integral whose integrand

is only stochastically square integrable. Here X ln is the step function defined by

X ln (t)≡

mn−1

∑k=0

X (tnk )X[tn

k ,tnk+1)

(t)

In particular, 65.14.21 is a local martingale.

Of course it would be very interesting to see what happens in the case where X ln→ X

in L2 ([0,T ]×Ω,H). Is it the case that convergence to∫ t

0R((

Z ◦ J−1)∗ (X))◦ JdW (65.14.22)

happens in some sense? Also, does the above stochastic integral even make sense? Firstof all, consider the question whether it makes sense. It would be nice to define a stoppingtime

τn ≡ inf{t : |X (t)|H > n}