118 7 A Primer in Stochastic Integration
quite a delicate task. The difficulties appear already in the case of Brownian
motion; we concentrate for the moment on the caseSt=Wt,where(Wt)t≥ 0
is a standard real-valued Brownian motion.
It was K. Itˆo’s fundamental insight [I 44] that the good idea isnot to proceed
in a pathwise way, i.e., not to consider eachω∈Ω separately. Instead one
should take a functional-analytic point of view applying a basic isometry of
Hilbert spaces. Consider the simple strategiesH=
∑n− 1
k=0fk^1 ]]Tk,Tk+1]] which
are bounded and of bounded support as elements ofL^2 (Ω×R+,P,P⊗λ),
wherePdenotes the predictableσ-algebra on Ω×R+andP⊗λthe product
measure ofPwith Lebesgue measureλonR+. This gives rise to the norm
‖H‖L (^2) (P⊗λ)=
(
E
[∫∞
0Hs^2 ds]) (^12)
. (7.7)
The crucial isometry is that thisL^2 (P⊗λ)-norm of theintegrandHequals
theL^2 (Ω,F∞,P)-norm of thestochastic integral(H·S)∞, i.e.,
‖H‖L^2 (Ω×R+,P,P⊗λ)=‖(H·W)∞‖L^2 (Ω,F,P)=(
E
[
(H·W)^2 ∞
])^12
, (7.8)
whereF∞denotes theσ-algebra generated by the Brownian motion (Wt)t≥ 0.
In fact, this isometry is essentially a formality: for simple integrandsHit
is a straightforward consequence of the definition of Brownian motion (see,
e.g., the beautiful introductory chapter of [RW 00]).
Having established (7.8) for the set of bounded simple integrands it now is
one more formal step to extend this isometry to the closures in the respective
Hilbert spacesL^2 (P⊗λ)andL^2 (P) respectively. For the former it follows
from the definition of the predictableσ-algebraPin Sect. 7.2 above that this
closure equals the entire spaceL^2 (Ω×R+,P,P⊗λ), i.e., the predictable
processHsuch that (7.7) remains finite. For the latter closure of the stochas-
tic integrals (H·S)∞inL^2 (P), it turns out that this is the hyperplane in
L^2 (Ω,F∞,P) formed by the random variablesfwithE[f] = 0. This amounts
to the martingale representation theorem 4.2.1 above.
We now define the processH·W=((H·W)t)t≥ 0 ,wherewehavetobe
careful as this involves uncountably manyt∈R+. This is done in the following
way. For generalH∈L^2 (Ω×R+,P,P⊗λ) we take a sequenceHnof simple
integrands,Hn∈L^2 (Ω×R+,P,P⊗λ)sothat‖H−Hn‖L (^2) (Ω×R+,P,P⊗λ)≤
4 −n−^1. By Doob’s maximal integrability [RW 00, Chap. 5, Theorem 70.2] we
get, for eachm, n∈N:
∥
∥
∥
∥supt≥ 0 ((H
m−Hn)·W)
t
∥
∥
∥
∥
L^2 (Ω,F,P)≤ 2 ‖((Hm−Hn)·W)∞‖L (^2) (Ω,F,P)
=2‖Hm−Hn‖L (^2) (Ω×R+,P,P⊗λ).
ThereforeP
[
supt≥ 0 ((Hn−Hn+1)·W)t> 2 −n]
≤ 2 −n. The Borel-Cantelli
lemma then implies that almost surely the sequence (Hn·W)t(ω)converges