7.3 Strategies, Semi-martingales and Stochastic Integration 119uniformly as a sequence of continuous functions onR+. Let us denote this
limit process by (H·W). Of course
∥
∥supt≥ 0 ((H−Hn)·W)t∥
∥
2 →0. The
identity (7.8) then shows, by passing to the limit asn→∞:
‖H (^1) ]] 0,t]]‖L^2 (P⊗λ)=‖(H·W)t‖L^2 (P).
Atthesametime(H·W), being the limit of theL^2 -martingales (Hn·W)t≥ 0 ,
also is a martingale bounded inL^2 (P).
We have taken some space to sketch these basic facts on stochastic inte-
gration, which can be found in much more detail in many beautiful textbooks
(e.g., [P 90], [RY 91], [RW 00]), as we believe that the isometric identity (7.8)
is the heart of the matter. Having clarified things for the case of Brownian
motionWit is essentially a matter of routine techniques to extend the degree
of generality.
To start we still restrict to the case of Brownian motionWbut now con-
sider predictable processesHsuch thatHis only locally inL^2 (P⊗λ). This
latter requirement is equivalent to the hypothesis
∫t
0 H
2
udu <∞a.s., for each
t<∞. In this case one can argue locally to define the stochastic integral
((H·S)t)t≥ 0 which then is a local martingale.
Passing to more general integrators than Brownian motionW consider a
real-valued martingaleS=(St)t≥ 0 which we first assume to beL^2 -bounded,
i.e., supt‖St‖L (^2) (Ω,Ft,P)<∞.
We then may define thequadratic variation measure d[S] on the pre-
dictableσ-algebraPby
d[S](]]τ, σ]] ) : =E
[
|Sτ−Sσ|^2]
(7.9)
for all pairs of finite stopping timesτ≤σandthenextendthismeasuretoP
by sigma-additivity. The measured[S] is the analogue of the measureP⊗λ
in the case of Brownian motionS=W, and we again obtain the isometric
identity
‖H‖L (^2) (d[S])=‖(H·S)∞‖L (^2) (P), (7.10)
for each bounded simple integrandHsuch that the left hand side is finite.
In fact, the identity (7.10) now simply is a reformulation of the definition
(7.9). As in the case of Brownian motion, identity (7.10) allows to extend
the stochastic integral from simple bounded integrands to general predictable
processesH with finiteL^2 (d[S])-norm. By localisation this notion can be
extended to the case of martingalesSwhich are locallyL^2 -bounded as well as
to integrandsH, which are locally inL^2 (d[S]). For the case of continuous local
martingalesSthis is already the natural degree of generality as acontinuous
local martingale is automatically locallyL^2 -bounded. Finally we indicate that
the theory may also be extended to the case ofRd-valued local martingales by
equippingRdwith its Euclidean norm|.|and using the above Hilbert space
techniques.