8.2 The Crucial Lemma 135Let us compare Lemma 8.2.3 with the previous literature. An important
theorem of J. M ́emin [M 80] states the following: if a sequence of stochastic in-
tegrals ((Hn·S)t≥ 0 )∞n=0on a given semi-martingaleSis Cauchy with respect
to the semi-martingale topology, then the limit exists (as a semi-martingale)
and is of the form (H·S)t≥ 0 for someS-integrable predictable processH.
This theorem will finally play an important role in proving Lemma 8.2.3;
but we still have a long way to go before we can apply it, as the assumptions
of Lemma 8.2.3 a priori do not tell us anything about the convergence of the
sequence of processes ((Hn·S)t≥ 0 )∞n=0.
Another line of results in the spirit of Lemma 8.2.3 assumes that the
processSis a (local) martingale. The arch-example is the theorem of Kunita-
Watanabe ([KW 67]; see also [P 90] or [Y 78a] or Chap. 7 above): suppose
thatSis a locallyL^2 -bounded martingale, that each (Hn·S)t≥ 0 is anL^2 -
bounded martingale, and that the sequence ((Hn·S)t≥ 0 )∞n=0is Cauchy in the
Hilbert space of square-integrable martingales (equivalently: that the sequence
of terminal values ((Hn·S)∞)∞n=0is Cauchy in the Hilbert spaceL^2 (Ω,F,P)).
Then the limit exists (as a square-integrable martingale) and it is of the form
(H·S)t≥ 0.
As the proof of this theorem is very simple and allows some insight into
the present theme, we sketch it (assuming, for notational simplicity, thatSis
R-valued): denote byd[S] the quadratic variation measure of the locallyL^2 -
bounded martingaleSas in (7.9), which defines a sigma-finite measure on the
σ-algebraPof predictable subsets ofR+×Ω. Denoting byL^2 (R+×Ω,P,d[S])
the corresponding Hilbert space, the stochastic integration theory is designed
in such a way that we have the isometric identity
‖H‖L^2 (R+×Ω,P,d[S])=‖(H·S)∞‖L^2 (Ω,F,P), (8.6)for each predictable processH, for which the left hand side of (8.6) is finite
(see Chap. 7 above).
Hence the assumption that ((Hn·S)t≥ 0 )∞n=0 is Cauchy in the Hilbert
space of square-integrable martingales is tantamount to the assumption that
(Hn)∞n=0is Cauchy inL^2 (R+×Ω,P,d[S]). Now, once more, the stochas-
tic integration theory is designed in a way thatL^2 (R+×Ω,P,d[S]) con-
sists precisely of theS-integrable, predictable processesHsuch thatH·Sis
anL^2 -bounded martingale. Hence by the completeness of the Hilbert space
L^2 (R+×Ω,P,d[S]) we can pass from the Cauchy-sequence (Hn)∞n=0to its
limitH∈L^2 (R+×Ω,P,d[S]). This finishes the sketch of the proof of the
Kunita-Watanabe theorem.
The above argument shows in a nice and transparent way how to deduce
from a completeness property of thespace of predictable integrandsHacom-
pleteness property of the correspondingspace of terminal results(H·S)∞
of stochastic integrals. In the context of the theorem of Kunita-Watanabe,
the functional analytic background for this argument is reduced to the — al-
most trivial — isometric identification of the two corresponding Hilbert spaces
in (8.6).